Find the Critical Points h(p)=(p-5)/(p^2+2)

$h (p) = \frac{p - 5}{p^{2} + 2}$

Find the derivative.

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Differentiate using the Quotient Rule which states that $\frac{d}{d p} [\frac{f (p)}{g (p)}]$ is $\frac{g (p) \frac{d}{d p} [f (p)] - f (p) \frac{d}{d p} [g (p)]}{{g (p)}^{2}}$ where $f (p) = p - 5$ and $g (p) = p^{2} + 2$ .

$\frac{(p^{2} + 2) \frac{d}{d p} [p - 5] - (p - 5) \frac{d}{d p} [p^{2} + 2]}{{(p^{2} + 2)}^{2}}$

Differentiate.

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By the Sum Rule, the derivative of $p - 5$ with respect to $p$ is $\frac{d}{d p} [p] + \frac{d}{d p} [- 5]$ .

$\frac{(p^{2} + 2) (\frac{d}{d p} [p] + \frac{d}{d p} [- 5]) - (p - 5) \frac{d}{d p} [p^{2} + 2]}{{(p^{2} + 2)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d p} [p^{n}]$ is $n p^{n - 1}$ where $n = 1$ .

$\frac{(p^{2} + 2) (1 + \frac{d}{d p} [- 5]) - (p - 5) \frac{d}{d p} [p^{2} + 2]}{{(p^{2} + 2)}^{2}}$

Since $- 5$ is constant with respect to $p$ , the derivative of $- 5$ with respect to $p$ is $0$ .

$\frac{(p^{2} + 2) (1 + 0) - (p - 5) \frac{d}{d p} [p^{2} + 2]}{{(p^{2} + 2)}^{2}}$

Simplify the expression.

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Add $1$ and $0$ .

$\frac{(p^{2} + 2) \cdot 1 - (p - 5) \frac{d}{d p} [p^{2} + 2]}{{(p^{2} + 2)}^{2}}$

Multiply $p^{2} + 2$ by $1$ .

$\frac{p^{2} + 2 - (p - 5) \frac{d}{d p} [p^{2} + 2]}{{(p^{2} + 2)}^{2}}$

By the Sum Rule, the derivative of $p^{2} + 2$ with respect to $p$ is $\frac{d}{d p} [p^{2}] + \frac{d}{d p} [2]$ .

$\frac{p^{2} + 2 - (p - 5) (\frac{d}{d p} [p^{2}] + \frac{d}{d p} [2])}{{(p^{2} + 2)}^{2}}$

Differentiate using the Power Rule which states that $\frac{d}{d p} [p^{n}]$ is $n p^{n - 1}$ where $n = 2$ .

$\frac{p^{2} + 2 - (p - 5) (2 p + \frac{d}{d p} [2])}{{(p^{2} + 2)}^{2}}$

Since $2$ is constant with respect to $p$ , the derivative of $2$ with respect to $p$ is $0$ .

$\frac{p^{2} + 2 - (p - 5) (2 p + 0)}{{(p^{2} + 2)}^{2}}$

Simplify the expression.

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Add $2 p$ and $0$ .

$\frac{p^{2} + 2 - (p - 5) (2 p)}{{(p^{2} + 2)}^{2}}$

Multiply $2$ by $- 1$ .

$\frac{p^{2} + 2 - 2 (p - 5) p}{{(p^{2} + 2)}^{2}}$

Simplify.

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Apply the distributive property.

$\frac{p^{2} + 2 + (- 2 p - 2 \cdot - 5) p}{{(p^{2} + 2)}^{2}}$

Apply the distributive property.

$\frac{p^{2} + 2 - 2 p \cdot p - 2 \cdot - 5 p}{{(p^{2} + 2)}^{2}}$

Simplify the numerator.

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Simplify each term.

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Multiply $p$ by $p$ by adding the exponents.

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Move $p$ .

$\frac{p^{2} + 2 - 2 (p \cdot p) - 2 \cdot - 5 p}{{(p^{2} + 2)}^{2}}$

Multiply $p$ by $p$ .

$\frac{p^{2} + 2 - 2 p^{2} - 2 \cdot - 5 p}{{(p^{2} + 2)}^{2}}$

Multiply $- 2$ by $- 5$ .

$\frac{p^{2} + 2 - 2 p^{2} + 10 p}{{(p^{2} + 2)}^{2}}$

Subtract $2 p^{2}$ from $p^{2}$ .

$\frac{- p^{2} + 2 + 10 p}{{(p^{2} + 2)}^{2}}$

Factor $- 1$ out of $- p^{2}$ .

$\frac{- (p^{2}) + 2 + 10 p}{{(p^{2} + 2)}^{2}}$

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- (p^{2}) - 1 (- 2) + 10 p}{{(p^{2} + 2)}^{2}}$

Factor $- 1$ out of $- (p^{2}) - 1 (- 2)$ .

$\frac{- (p^{2} - 2) + 10 p}{{(p^{2} + 2)}^{2}}$

Factor $- 1$ out of $10 p$ .

$\frac{- (p^{2} - 2) - (- 10 p)}{{(p^{2} + 2)}^{2}}$

Factor $- 1$ out of $- (p^{2} - 2) - (- 10 p)$ .

$\frac{- (p^{2} - 2 - 10 p)}{{(p^{2} + 2)}^{2}}$

Rewrite $- (p^{2} - 2 - 10 p)$ as $- 1 (p^{2} - 2 - 10 p)$ .

$\frac{- 1 (p^{2} - 2 - 10 p)}{{(p^{2} + 2)}^{2}}$

Move the negative in front of the fraction.

$- \frac{p^{2} - 2 - 10 p}{{(p^{2} + 2)}^{2}}$

Set the derivative equal to $0$ .

$- \frac{p^{2} - 2 - 10 p}{{(p^{2} + 2)}^{2}} = 0$

Solve for $p$ .

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Reorder terms.

$- \frac{p^{2} - 10 p - 2}{{(p^{2} + 2)}^{2}} = 0$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

${(p^{2} + 2)}^{2}, 1$

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factors for $p^{2} + 2$ are $(p^{2} + 2) \cdot (p^{2} + 2)$ , which is $p^{2} + 2$ multiplied by itself $2$ times.

$(p^{2} + 2) = (p^{2} + 2) \cdot (p^{2} + 2)$

$(p^{2} + 2)$ occurs $2$ times.

The LCM of ${(p^{2} + 2)}^{2}$ is the result of multiplying all factors the greatest number of times they occur in either term.

${(p^{2} + 2)}^{2}$

Multiply each term by ${(p^{2} + 2)}^{2}$ and simplify.

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Multiply each term in $- \frac{p^{2} - 10 p - 2}{{(p^{2} + 2)}^{2}} = 0$ by ${(p^{2} + 2)}^{2}$ in order to remove all the denominators from the equation.

$- \frac{p^{2} - 10 p - 2}{{(p^{2} + 2)}^{2}} \cdot {(p^{2} + 2)}^{2} = 0 \cdot {(p^{2} + 2)}^{2}$

Simplify $- \frac{p^{2} - 10 p - 2}{{(p^{2} + 2)}^{2}} \cdot {(p^{2} + 2)}^{2}$ .

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Cancel the common factor of ${(p^{2} + 2)}^{2}$ .

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Move the leading negative in $- \frac{p^{2} - 10 p - 2}{{(p^{2} + 2)}^{2}}$ into the numerator.

$\frac{- (p^{2} - 10 p - 2)}{{(p^{2} + 2)}^{2}} \cdot {(p^{2} + 2)}^{2} = 0 \cdot {(p^{2} + 2)}^{2}$

Cancel the common factor.

$\frac{- (p^{2} - 10 p - 2)}{{(p^{2} + 2)}^{2}} \cdot {(p^{2} + 2)}^{2} = 0 \cdot {(p^{2} + 2)}^{2}$

Rewrite the expression.

$- (p^{2} - 10 p - 2) = 0 \cdot {(p^{2} + 2)}^{2}$

Apply the distributive property.

$- p^{2} - (- 10 p) - - 2 = 0 \cdot {(p^{2} + 2)}^{2}$

Simplify.

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Multiply $- 10$ by $- 1$ .

$- p^{2} + 10 p - - 2 = 0 \cdot {(p^{2} + 2)}^{2}$

Multiply $- 1$ by $- 2$ .

$- p^{2} + 10 p + 2 = 0 \cdot {(p^{2} + 2)}^{2}$

Simplify $0 \cdot {(p^{2} + 2)}^{2}$ .

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Rewrite ${(p^{2} + 2)}^{2}$ as $(p^{2} + 2) (p^{2} + 2)$ .

$- p^{2} + 10 p + 2 = 0 \cdot ((p^{2} + 2) (p^{2} + 2))$

Expand $(p^{2} + 2) (p^{2} + 2)$ using the FOIL Method.

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Apply the distributive property.

$- p^{2} + 10 p + 2 = 0 \cdot (p^{2} (p^{2} + 2) + 2 (p^{2} + 2))$

Apply the distributive property.

$- p^{2} + 10 p + 2 = 0 \cdot (p^{2} p^{2} + p^{2} \cdot 2 + 2 (p^{2} + 2))$

Apply the distributive property.

$- p^{2} + 10 p + 2 = 0 \cdot (p^{2} p^{2} + p^{2} \cdot 2 + 2 p^{2} + 2 \cdot 2)$

Simplify and combine like terms.

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Simplify each term.

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Multiply $p^{2}$ by $p^{2}$ by adding the exponents.

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Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- p^{2} + 10 p + 2 = 0 \cdot (p^{2 + 2} + p^{2} \cdot 2 + 2 p^{2} + 2 \cdot 2)$

Add $2$ and $2$ .

$- p^{2} + 10 p + 2 = 0 \cdot (p^{4} + p^{2} \cdot 2 + 2 p^{2} + 2 \cdot 2)$

Move $2$ to the left of $p^{2}$ .

$- p^{2} + 10 p + 2 = 0 \cdot (p^{4} + 2 \cdot p^{2} + 2 p^{2} + 2 \cdot 2)$

Multiply $2$ by $2$ .

$- p^{2} + 10 p + 2 = 0 \cdot (p^{4} + 2 p^{2} + 2 p^{2} + 4)$

Add $2 p^{2}$ and $2 p^{2}$ .

$- p^{2} + 10 p + 2 = 0 \cdot (p^{4} + 4 p^{2} + 4)$

Multiply $0$ by $p^{4} + 4 p^{2} + 4$ .

$- p^{2} + 10 p + 2 = 0$

Solve the equation.

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Factor $- 1$ out of $- p^{2} + 10 p + 2$ .

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Factor $- 1$ out of $- p^{2}$ .

$- (p^{2}) + 10 p + 2 = 0$

Factor $- 1$ out of $10 p$ .

$- (p^{2}) - (- 10 p) + 2 = 0$

Rewrite $2$ as $- 1 (- 2)$ .

$- (p^{2}) - (- 10 p) - 1 \cdot - 2 = 0$

Factor $- 1$ out of $- (p^{2}) - (- 10 p)$ .

$- (p^{2} - 10 p) - 1 \cdot - 2 = 0$

Factor $- 1$ out of $- (p^{2} - 10 p) - 1 (- 2)$ .

$- (p^{2} - 10 p - 2) = 0$

Multiply each term in $- (p^{2} - 10 p - 2) = 0$ by $- 1$

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Multiply each term in $- (p^{2} - 10 p - 2) = 0$ by $- 1$ .

$- (p^{2} - 10 p - 2) \cdot - 1 = 0 \cdot - 1$

Simplify $- (p^{2} - 10 p - 2) \cdot - 1$ .

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Apply the distributive property.

$(- p^{2} - (- 10 p) - - 2) \cdot - 1 = 0 \cdot - 1$

Simplify.

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Multiply $- 10$ by $- 1$ .

$(- p^{2} + 10 p - - 2) \cdot - 1 = 0 \cdot - 1$

Multiply $- 1$ by $- 2$ .

$(- p^{2} + 10 p + 2) \cdot - 1 = 0 \cdot - 1$

Apply the distributive property.

$- p^{2} \cdot - 1 + 10 p \cdot - 1 + 2 \cdot - 1 = 0 \cdot - 1$

Simplify.

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Multiply $- p^{2} \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 p^{2} + 10 p \cdot - 1 + 2 \cdot - 1 = 0 \cdot - 1$

Multiply $p^{2}$ by $1$ .

$p^{2} + 10 p \cdot - 1 + 2 \cdot - 1 = 0 \cdot - 1$

Multiply $- 1$ by $10$ .

$p^{2} - 10 p + 2 \cdot - 1 = 0 \cdot - 1$

Multiply $2$ by $- 1$ .

$p^{2} - 10 p - 2 = 0 \cdot - 1$

Multiply $0$ by $- 1$ .

$p^{2} - 10 p - 2 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 10$ , and $c = - 2$ into the quadratic formula and solve for $p$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot - 2)}}{2 \cdot 1}$

Simplify.

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Simplify the numerator.

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Raise $- 10$ to the power of $2$ .

$p = \frac{10 \pm \sqrt{100 - 4 \cdot (1 \cdot - 2)}}{2 \cdot 1}$

Multiply $- 2$ by $1$ .

$p = \frac{10 \pm \sqrt{100 - 4 \cdot - 2}}{2 \cdot 1}$

Multiply $- 4$ by $- 2$ .

$p = \frac{10 \pm \sqrt{100 + 8}}{2 \cdot 1}$

Add $100$ and $8$ .

$p = \frac{10 \pm \sqrt{108}}{2 \cdot 1}$

Rewrite $108$ as $6^{2} \cdot 3$ .

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Factor $36$ out of $108$ .

$p = \frac{10 \pm \sqrt{36 (3)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$p = \frac{10 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$p = \frac{10 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$p = \frac{10 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{10 \pm 6 \sqrt{3}}{2}$ .

$p = 5 \pm 3 \sqrt{3}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $- 10$ to the power of $2$ .

$p = \frac{10 \pm \sqrt{100 - 4 \cdot (1 \cdot - 2)}}{2 \cdot 1}$

Multiply $- 2$ by $1$ .

$p = \frac{10 \pm \sqrt{100 - 4 \cdot - 2}}{2 \cdot 1}$

Multiply $- 4$ by $- 2$ .

$p = \frac{10 \pm \sqrt{100 + 8}}{2 \cdot 1}$

Add $100$ and $8$ .

$p = \frac{10 \pm \sqrt{108}}{2 \cdot 1}$

Rewrite $108$ as $6^{2} \cdot 3$ .

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Factor $36$ out of $108$ .

$p = \frac{10 \pm \sqrt{36 (3)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$p = \frac{10 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$p = \frac{10 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$p = \frac{10 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{10 \pm 6 \sqrt{3}}{2}$ .

$p = 5 \pm 3 \sqrt{3}$

Change the $\pm$ to $+$ .

$p = 5 + 3 \sqrt{3}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $- 10$ to the power of $2$ .

$p = \frac{10 \pm \sqrt{100 - 4 \cdot (1 \cdot - 2)}}{2 \cdot 1}$

Multiply $- 2$ by $1$ .

$p = \frac{10 \pm \sqrt{100 - 4 \cdot - 2}}{2 \cdot 1}$

Multiply $- 4$ by $- 2$ .

$p = \frac{10 \pm \sqrt{100 + 8}}{2 \cdot 1}$

Add $100$ and $8$ .

$p = \frac{10 \pm \sqrt{108}}{2 \cdot 1}$

Rewrite $108$ as $6^{2} \cdot 3$ .

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Factor $36$ out of $108$ .

$p = \frac{10 \pm \sqrt{36 (3)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$p = \frac{10 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$p = \frac{10 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$p = \frac{10 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{10 \pm 6 \sqrt{3}}{2}$ .

$p = 5 \pm 3 \sqrt{3}$

Change the $\pm$ to $-$ .

$p = 5 - 3 \sqrt{3}$

The final answer is the combination of both solutions.

$p = 5 + 3 \sqrt{3}, 5 - 3 \sqrt{3}$

Substitute the values of $x$ which cause the derivative to be $0$ into the original function.

$h (5 + 3 \sqrt{3}) = \frac{(5 + 3 \sqrt{3}) - 5}{{(5 + 3 \sqrt{3})}^{2} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Evaluate.

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Simplify the numerator.

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Subtract $5$ from $5$ .

$h (5 + 3 \sqrt{3}) = \frac{0 + 3 \sqrt{3}}{{(5 + 3 \sqrt{3})}^{2} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Add $0$ and $3 \sqrt{3}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{{(5 + 3 \sqrt{3})}^{2} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{{(5 + 3 \sqrt{3})}^{2} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Simplify the denominator.

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Rewrite ${(5 + 3 \sqrt{3})}^{2}$ as $(5 + 3 \sqrt{3}) (5 + 3 \sqrt{3})$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{(5 + 3 \sqrt{3}) (5 + 3 \sqrt{3}) + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Expand $(5 + 3 \sqrt{3}) (5 + 3 \sqrt{3})$ using the FOIL Method.

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Apply the distributive property.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{5 (5 + 3 \sqrt{3}) + 3 \sqrt{3} (5 + 3 \sqrt{3}) + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Apply the distributive property.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{5 \cdot 5 + 5 (3 \sqrt{3}) + 3 \sqrt{3} (5 + 3 \sqrt{3}) + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Apply the distributive property.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{5 \cdot 5 + 5 (3 \sqrt{3}) + 3 \sqrt{3} \cdot 5 + 3 \sqrt{3} (3 \sqrt{3}) + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{5 \cdot 5 + 5 (3 \sqrt{3}) + 3 \sqrt{3} \cdot 5 + 3 \sqrt{3} (3 \sqrt{3}) + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Simplify and combine like terms.

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Simplify each term.

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Multiply $5$ by $5$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 5 (3 \sqrt{3}) + 3 \sqrt{3} \cdot 5 + 3 \sqrt{3} (3 \sqrt{3}) + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Multiply $3$ by $5$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 3 \sqrt{3} \cdot 5 + 3 \sqrt{3} (3 \sqrt{3}) + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Multiply $5$ by $3$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 3 \sqrt{3} (3 \sqrt{3}) + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Multiply $3 \sqrt{3} (3 \sqrt{3})$ .

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Multiply $3$ by $3$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 \sqrt{3} \sqrt{3} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Raise $\sqrt{3}$ to the power of $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 (\sqrt{3} \sqrt{3}) + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Raise $\sqrt{3}$ to the power of $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 (\sqrt{3} \sqrt{3}) + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 {\sqrt{3}}^{1 + 1} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Add $1$ and $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 {\sqrt{3}}^{2} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 {\sqrt{3}}^{2} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 {(3^{\frac{1}{2}})}^{2} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 \cdot 3^{\frac{1}{2} \cdot 2} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Combine $\frac{1}{2}$ and $2$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 \cdot 3^{\frac{2}{2}} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 \cdot 3^{\frac{2}{2}} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Divide $1$ by $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 \cdot 3 + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 \cdot 3 + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Evaluate the exponent.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 \cdot 3 + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 9 \cdot 3 + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Multiply $9$ by $3$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 27 + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{25 + 15 \sqrt{3} + 15 \sqrt{3} + 27 + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Add $25$ and $27$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{52 + 15 \sqrt{3} + 15 \sqrt{3} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Add $15 \sqrt{3}$ and $15 \sqrt{3}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{52 + 30 \sqrt{3} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{52 + 30 \sqrt{3} + 2}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Add $52$ and $2$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{54 + 30 \sqrt{3}}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{54 + 30 \sqrt{3}}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Cancel the common factor of $3$ and $54 + 30 \sqrt{3}$ .

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Factor $3$ out of $3 \sqrt{3}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 (\sqrt{3})}{54 + 30 \sqrt{3}}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Cancel the common factors.

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Factor $3$ out of $54$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{3 \cdot 18 + 30 \sqrt{3}}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Factor $3$ out of $30 \sqrt{3}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{3 \cdot 18 + 3 (10 \sqrt{3})}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Factor $3$ out of $3 (18) + 3 (10 \sqrt{3})$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{3 (18 + 10 \sqrt{3})}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Cancel the common factor.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3}}{3 (18 + 10 \sqrt{3})}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Rewrite the expression.

$h (5 + 3 \sqrt{3}) = \frac{\sqrt{3}}{18 + 10 \sqrt{3}}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{\sqrt{3}}{18 + 10 \sqrt{3}}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{\sqrt{3}}{18 + 10 \sqrt{3}}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Multiply $\frac{\sqrt{3}}{18 + 10 \sqrt{3}}$ by $\frac{18 - 10 \sqrt{3}}{18 - 10 \sqrt{3}}$ .

$h (5 + 3 \sqrt{3}) = \frac{\sqrt{3}}{18 + 10 \sqrt{3}} \cdot \frac{18 - 10 \sqrt{3}}{18 - 10 \sqrt{3}}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Multiply $\frac{\sqrt{3}}{18 + 10 \sqrt{3}}$ and $\frac{18 - 10 \sqrt{3}}{18 - 10 \sqrt{3}}$ .

$h (5 + 3 \sqrt{3}) = \frac{\sqrt{3} (18 - 10 \sqrt{3})}{(18 + 10 \sqrt{3}) (18 - 10 \sqrt{3})}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Expand the denominator using the FOIL method.

$h (5 + 3 \sqrt{3}) = \frac{\sqrt{3} (18 - 10 \sqrt{3})}{324 - 180 \sqrt{3} + 180 \sqrt{3} - 100 {\sqrt{3}}^{2}}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Simplify.

$h (5 + 3 \sqrt{3}) = \frac{\sqrt{3} (18 - 10 \sqrt{3})}{24}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Cancel the common factor of $18 - 10 \sqrt{3}$ and $24$ .

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Factor $2$ out of $\sqrt{3} (18 - 10 \sqrt{3})$ .

$h (5 + 3 \sqrt{3}) = \frac{2 (\sqrt{3} (9 - 5 \sqrt{3}))}{24}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Cancel the common factors.

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Factor $2$ out of $24$ .

$h (5 + 3 \sqrt{3}) = \frac{2 (\sqrt{3} (9 - 5 \sqrt{3}))}{2 (12)}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Cancel the common factor.

$h (5 + 3 \sqrt{3}) = \frac{2 (\sqrt{3} (9 - 5 \sqrt{3}))}{2 \cdot 12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Rewrite the expression.

$h (5 + 3 \sqrt{3}) = \frac{\sqrt{3} (9 - 5 \sqrt{3})}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{\sqrt{3} (9 - 5 \sqrt{3})}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{\sqrt{3} (9 - 5 \sqrt{3})}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Apply the distributive property.

$h (5 + 3 \sqrt{3}) = \frac{\sqrt{3} \cdot 9 + \sqrt{3} (- 5 \sqrt{3})}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Move $9$ to the left of $\sqrt{3}$ .

$h (5 + 3 \sqrt{3}) = \frac{9 \cdot \sqrt{3} + \sqrt{3} (- 5 \sqrt{3})}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Multiply $\sqrt{3} (- 5 \sqrt{3})$ .

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Raise $\sqrt{3}$ to the power of $1$ .

$h (5 + 3 \sqrt{3}) = \frac{9 \cdot \sqrt{3} - 5 (\sqrt{3} \sqrt{3})}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Raise $\sqrt{3}$ to the power of $1$ .

$h (5 + 3 \sqrt{3}) = \frac{9 \cdot \sqrt{3} - 5 (\sqrt{3} \sqrt{3})}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$h (5 + 3 \sqrt{3}) = \frac{9 \cdot \sqrt{3} - 5 {\sqrt{3}}^{1 + 1}}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Add $1$ and $1$ .

$h (5 + 3 \sqrt{3}) = \frac{9 \cdot \sqrt{3} - 5 {\sqrt{3}}^{2}}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{9 \cdot \sqrt{3} - 5 {\sqrt{3}}^{2}}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Simplify each term.

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Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$h (5 + 3 \sqrt{3}) = \frac{9 \sqrt{3} - 5 {(3^{\frac{1}{2}})}^{2}}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$h (5 + 3 \sqrt{3}) = \frac{9 \sqrt{3} - 5 \cdot 3^{\frac{1}{2} \cdot 2}}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Combine $\frac{1}{2}$ and $2$ .

$h (5 + 3 \sqrt{3}) = \frac{9 \sqrt{3} - 5 \cdot 3^{\frac{2}{2}}}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$h (5 + 3 \sqrt{3}) = \frac{9 \sqrt{3} - 5 \cdot 3^{\frac{2}{2}}}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Divide $1$ by $1$ .

$h (5 + 3 \sqrt{3}) = \frac{9 \sqrt{3} - 5 \cdot 3}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{9 \sqrt{3} - 5 \cdot 3}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Evaluate the exponent.

$h (5 + 3 \sqrt{3}) = \frac{9 \sqrt{3} - 5 \cdot 3}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{9 \sqrt{3} - 5 \cdot 3}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Multiply $- 5$ by $3$ .

$h (5 + 3 \sqrt{3}) = \frac{9 \sqrt{3} - 15}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{9 \sqrt{3} - 15}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Cancel the common factor of $9 \sqrt{3} - 15$ and $12$ .

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Factor $3$ out of $9 \sqrt{3}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 (3 \sqrt{3}) - 15}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Factor $3$ out of $- 15$ .

$h (5 + 3 \sqrt{3}) = \frac{3 (3 \sqrt{3}) + 3 (- 5)}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Factor $3$ out of $3 (3 \sqrt{3}) + 3 (- 5)$ .

$h (5 + 3 \sqrt{3}) = \frac{3 (3 \sqrt{3} - 5)}{12}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Cancel the common factors.

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Factor $3$ out of $12$ .

$h (5 + 3 \sqrt{3}) = \frac{3 (3 \sqrt{3} - 5)}{3 (4)}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Cancel the common factor.

$h (5 + 3 \sqrt{3}) = \frac{3 (3 \sqrt{3} - 5)}{3 \cdot 4}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Rewrite the expression.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{(5 - 3 \sqrt{3}) - 5}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Simplify the numerator.

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Subtract $5$ from $5$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{0 - 3 \sqrt{3}}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Subtract $3 \sqrt{3}$ from $0$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{{(5 - 3 \sqrt{3})}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{{(5 - 3 \sqrt{3})}^{2} + 2}$

Simplify the denominator.

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Rewrite ${(5 - 3 \sqrt{3})}^{2}$ as $(5 - 3 \sqrt{3}) (5 - 3 \sqrt{3})$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{(5 - 3 \sqrt{3}) (5 - 3 \sqrt{3}) + 2}$

Expand $(5 - 3 \sqrt{3}) (5 - 3 \sqrt{3})$ using the FOIL Method.

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Apply the distributive property.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{5 (5 - 3 \sqrt{3}) - 3 \sqrt{3} (5 - 3 \sqrt{3}) + 2}$

Apply the distributive property.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{5 \cdot 5 + 5 (- 3 \sqrt{3}) - 3 \sqrt{3} (5 - 3 \sqrt{3}) + 2}$

Apply the distributive property.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{5 \cdot 5 + 5 (- 3 \sqrt{3}) - 3 \sqrt{3} \cdot 5 - 3 \sqrt{3} (- 3 \sqrt{3}) + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{5 \cdot 5 + 5 (- 3 \sqrt{3}) - 3 \sqrt{3} \cdot 5 - 3 \sqrt{3} (- 3 \sqrt{3}) + 2}$

Simplify and combine like terms.

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Simplify each term.

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Multiply $5$ by $5$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 + 5 (- 3 \sqrt{3}) - 3 \sqrt{3} \cdot 5 - 3 \sqrt{3} (- 3 \sqrt{3}) + 2}$

Multiply $- 3$ by $5$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 3 \sqrt{3} \cdot 5 - 3 \sqrt{3} (- 3 \sqrt{3}) + 2}$

Multiply $5$ by $- 3$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} - 3 \sqrt{3} (- 3 \sqrt{3}) + 2}$

Multiply $- 3 \sqrt{3} (- 3 \sqrt{3})$ .

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Multiply $- 3$ by $- 3$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 \sqrt{3} \sqrt{3} + 2}$

Raise $\sqrt{3}$ to the power of $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 (\sqrt{3} \sqrt{3}) + 2}$

Raise $\sqrt{3}$ to the power of $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 (\sqrt{3} \sqrt{3}) + 2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 {\sqrt{3}}^{1 + 1} + 2}$

Add $1$ and $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 {\sqrt{3}}^{2} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 {\sqrt{3}}^{2} + 2}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 {(3^{\frac{1}{2}})}^{2} + 2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 \cdot 3^{\frac{1}{2} \cdot 2} + 2}$

Combine $\frac{1}{2}$ and $2$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 \cdot 3^{\frac{2}{2}} + 2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 \cdot 3^{\frac{2}{2}} + 2}$

Divide $1$ by $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 \cdot 3 + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 \cdot 3 + 2}$

Evaluate the exponent.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 \cdot 3 + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 9 \cdot 3 + 2}$

Multiply $9$ by $3$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 27 + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{25 - 15 \sqrt{3} - 15 \sqrt{3} + 27 + 2}$

Add $25$ and $27$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{52 - 15 \sqrt{3} - 15 \sqrt{3} + 2}$

Subtract $15 \sqrt{3}$ from $- 15 \sqrt{3}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{52 - 30 \sqrt{3} + 2}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{52 - 30 \sqrt{3} + 2}$

Add $52$ and $2$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{54 - 30 \sqrt{3}}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- 3 \sqrt{3}}{54 - 30 \sqrt{3}}$

Cancel the common factor of $- 3$ and $54 - 30 \sqrt{3}$ .

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Factor $3$ out of $- 3 \sqrt{3}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{3 (- \sqrt{3})}{54 - 30 \sqrt{3}}$

Cancel the common factors.

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Factor $3$ out of $54$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{3 (- \sqrt{3})}{3 (18) - 30 \sqrt{3}}$

Factor $3$ out of $- 30 \sqrt{3}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{3 (- \sqrt{3})}{3 (18) + 3 (- 10 \sqrt{3})}$

Factor $3$ out of $3 (18) + 3 (- 10 \sqrt{3})$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{3 (- \sqrt{3})}{3 (18 - 10 \sqrt{3})}$

Cancel the common factor.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{3 (- \sqrt{3})}{3 (18 - 10 \sqrt{3})}$

Rewrite the expression.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- \sqrt{3}}{18 - 10 \sqrt{3}}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- \sqrt{3}}{18 - 10 \sqrt{3}}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- \sqrt{3}}{18 - 10 \sqrt{3}}$

Multiply $\frac{- \sqrt{3}}{18 - 10 \sqrt{3}}$ by $\frac{18 + 10 \sqrt{3}}{18 + 10 \sqrt{3}}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- \sqrt{3}}{18 - 10 \sqrt{3}} \cdot \frac{18 + 10 \sqrt{3}}{18 + 10 \sqrt{3}}$

Multiply $\frac{- \sqrt{3}}{18 - 10 \sqrt{3}}$ and $\frac{18 + 10 \sqrt{3}}{18 + 10 \sqrt{3}}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- \sqrt{3} (18 + 10 \sqrt{3})}{(18 - 10 \sqrt{3}) (18 + 10 \sqrt{3})}$

Expand the denominator using the FOIL method.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- \sqrt{3} (18 + 10 \sqrt{3})}{324 + 180 \sqrt{3} - 180 \sqrt{3} - 100 {\sqrt{3}}^{2}}$

Simplify.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- \sqrt{3} (18 + 10 \sqrt{3})}{24}$

Cancel the common factor of $18 + 10 \sqrt{3}$ and $24$ .

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Factor $2$ out of $- \sqrt{3} (18 + 10 \sqrt{3})$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{2 (- \sqrt{3} (9 + 5 \sqrt{3}))}{24}$

Cancel the common factors.

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Factor $2$ out of $24$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{2 (- \sqrt{3} (9 + 5 \sqrt{3}))}{2 (12)}$

Cancel the common factor.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{2 (- \sqrt{3} (9 + 5 \sqrt{3}))}{2 \cdot 12}$

Rewrite the expression.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- \sqrt{3} (9 + 5 \sqrt{3})}{12}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- \sqrt{3} (9 + 5 \sqrt{3})}{12}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- \sqrt{3} (9 + 5 \sqrt{3})}{12}$

Group $9 + 5 \sqrt{3}$ and $\sqrt{3}$ together.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- ((9 + 5 \sqrt{3}) \sqrt{3})}{12}$

Apply the distributive property.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 \sqrt{3} \sqrt{3})}{12}$

Multiply $5 \sqrt{3} \sqrt{3}$ .

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Raise $\sqrt{3}$ to the power of $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 (\sqrt{3} \sqrt{3}))}{12}$

Raise $\sqrt{3}$ to the power of $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 (\sqrt{3} \sqrt{3}))}{12}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 {\sqrt{3}}^{1 + 1})}{12}$

Add $1$ and $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 {\sqrt{3}}^{2})}{12}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 {\sqrt{3}}^{2})}{12}$

Simplify each term.

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Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 {(3^{\frac{1}{2}})}^{2})}{12}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 \cdot 3^{\frac{1}{2} \cdot 2})}{12}$

Combine $\frac{1}{2}$ and $2$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 \cdot 3^{\frac{2}{2}})}{12}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 \cdot 3^{\frac{2}{2}})}{12}$

Divide $1$ by $1$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 \cdot 3)}{12}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 \cdot 3)}{12}$

Evaluate the exponent.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 \cdot 3)}{12}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 5 \cdot 3)}{12}$

Multiply $5$ by $3$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 15)}{12}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (9 \sqrt{3} + 15)}{12}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor of $9 \sqrt{3} + 15$ and $12$ .

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Factor $3$ out of $- (9 \sqrt{3} + 15)$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{3 (- (3 \sqrt{3} + 5))}{12}$

Cancel the common factors.

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Factor $3$ out of $12$ .

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{3 (- (3 \sqrt{3} + 5))}{3 (4)}$

Cancel the common factor.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{3 (- (3 \sqrt{3} + 5))}{3 \cdot 4}$

Rewrite the expression.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (3 \sqrt{3} + 5)}{4}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (3 \sqrt{3} + 5)}{4}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = \frac{- (3 \sqrt{3} + 5)}{4}$

Move the negative in front of the fraction.

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = - \frac{3 \sqrt{3} + 5}{4}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = - \frac{3 \sqrt{3} + 5}{4}$

$h (5 + 3 \sqrt{3}) = \frac{3 \sqrt{3} - 5}{4}$

$h (5 - 3 \sqrt{3}) = - \frac{3 \sqrt{3} + 5}{4}$

Find the domain of the derivative.

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Set the denominator in $\frac{p^{2} - 2 - 10 p}{{(p^{2} + 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(p^{2} + 2)}^{2} = 0$

Solve for $p$ .

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Set the $p^{2} + 2$ equal to $0$ .

$p^{2} + 2 = 0$

Solve for $p$ .

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Subtract $2$ from both sides of the equation.

$p^{2} = - 2$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$p = \pm \sqrt{- 2}$

The complete solution is the result of both the positive and negative portions of the solution.

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Simplify the right side of the equation.

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Rewrite $- 2$ as $- 1 (2)$ .

$p = \pm \sqrt{- 1 \cdot 2}$

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$p = \pm \sqrt{- 1} \cdot \sqrt{2}$

Rewrite $\sqrt{- 1}$ as $i$ .

$p = \pm i \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

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First, use the positive value of the $\pm$ to find the first solution.

$p = i \sqrt{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$p = - i \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$p = i \sqrt{2}, - i \sqrt{2}$

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${p | p \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${p | p \in ℝ}$

Since there are no values of $p$ where the derivative is undefined, there are no additional critical points.

$(5 + 3 \sqrt{3}, \frac{3 \sqrt{3} - 5}{4}), (5 - 3 \sqrt{3}, - \frac{3 \sqrt{3} + 5}{4})$

Find the Critical Points h(p)=(p-5)/(p^2+2)

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