Algebra Examples

$\frac{{(x^{2} - 4)}^{4}}{(x^{2} - 4 x + 4) \sqrt{x + 2}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

$\frac{{(x^{2} - 2^{2})}^{4}}{(x^{2} - 4 x + 4) \sqrt{x + 2}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{{((x + 2) (x - 2))}^{4}}{(x^{2} - 4 x + 4) \sqrt{x + 2}}$

Step 1.3

Apply the product rule to $(x + 2) (x - 2)$ .

$\frac{{(x + 2)}^{4} {(x - 2)}^{4}}{(x^{2} - 4 x + 4) \sqrt{x + 2}}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $4$ as $2^{2}$ .

$\frac{{(x + 2)}^{4} {(x - 2)}^{4}}{(x^{2} - 4 x + 2^{2}) \sqrt{x + 2}}$

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 2.3

Rewrite the polynomial.

$\frac{{(x + 2)}^{4} {(x - 2)}^{4}}{(x^{2} - 2 \cdot x \cdot 2 + 2^{2}) \sqrt{x + 2}}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

$\frac{{(x + 2)}^{4} {(x - 2)}^{4}}{{(x - 2)}^{2} \sqrt{x + 2}}$

Step 3

Cancel the common factor of ${(x - 2)}^{4}$ and ${(x - 2)}^{2}$ .

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Step 3.1

Factor ${(x - 2)}^{2}$ out of ${(x + 2)}^{4} {(x - 2)}^{4}$ .

$\frac{{(x - 2)}^{2} ({(x + 2)}^{4} {(x - 2)}^{2})}{{(x - 2)}^{2} \sqrt{x + 2}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor ${(x - 2)}^{2}$ out of ${(x - 2)}^{2} \sqrt{x + 2}$ .

$\frac{{(x - 2)}^{2} ({(x + 2)}^{4} {(x - 2)}^{2})}{{(x - 2)}^{2} (\sqrt{x + 2})}$

Step 3.2.2

Cancel the common factor.

$\frac{{(x - 2)}^{2} ({(x + 2)}^{4} {(x - 2)}^{2})}{{(x - 2)}^{2} \sqrt{x + 2}}$

Step 3.2.3

Rewrite the expression.

$\frac{{(x + 2)}^{4} {(x - 2)}^{2}}{\sqrt{x + 2}}$

Step 4

Multiply $\frac{{(x + 2)}^{4} {(x - 2)}^{2}}{\sqrt{x + 2}}$ by $\frac{\sqrt{x + 2}}{\sqrt{x + 2}}$ .

$\frac{{(x + 2)}^{4} {(x - 2)}^{2}}{\sqrt{x + 2}} \cdot \frac{\sqrt{x + 2}}{\sqrt{x + 2}}$

Step 5

Simplify terms.

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Step 5.1

Combine and simplify the denominator.

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Step 5.1.1

Multiply $\frac{{(x + 2)}^{4} {(x - 2)}^{2}}{\sqrt{x + 2}}$ by $\frac{\sqrt{x + 2}}{\sqrt{x + 2}}$ .

$\frac{{(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}}{\sqrt{x + 2} \sqrt{x + 2}}$

Step 5.1.2

Raise $\sqrt{x + 2}$ to the power of $1$ .

$\frac{{(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}}{{\sqrt{x + 2}}^{1} \sqrt{x + 2}}$

Step 5.1.3

Raise $\sqrt{x + 2}$ to the power of $1$ .

$\frac{{(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}}{{\sqrt{x + 2}}^{1} {\sqrt{x + 2}}^{1}}$

Step 5.1.4

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

$\frac{{(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}}{{\sqrt{x + 2}}^{1 + 1}}$

Step 5.1.5

Add $1$ and $1$ .

$\frac{{(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}}{{\sqrt{x + 2}}^{2}}$

Step 5.1.6

Rewrite ${\sqrt{x + 2}}^{2}$ as $x + 2$ .

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Step 5.1.6.1

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

$\frac{{(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}}{{({(x + 2)}^{\frac{1}{2}})}^{2}}$

Step 5.1.6.2

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

$\frac{{(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}}{{(x + 2)}^{\frac{1}{2} \cdot 2}}$

Step 5.1.6.3

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

$\frac{{(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}}{{(x + 2)}^{\frac{2}{2}}}$

Step 5.1.6.4

Cancel the common factor of $2$ .

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Step 5.1.6.4.1

Cancel the common factor.

$\frac{{(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}}{{(x + 2)}^{\frac{2}{2}}}$

Step 5.1.6.4.2

Rewrite the expression.

$\frac{{(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}}{{(x + 2)}^{1}}$

Step 5.1.6.5

Simplify.

$\frac{{(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}}{x + 2}$

Step 5.2

Cancel the common factor of ${(x + 2)}^{4}$ and $x + 2$ .

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Step 5.2.1

Factor $x + 2$ out of ${(x + 2)}^{4} {(x - 2)}^{2} \sqrt{x + 2}$ .

$\frac{(x + 2) ({(x + 2)}^{3} {(x - 2)}^{2} \sqrt{x + 2})}{x + 2}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Multiply by $1$ .

$\frac{(x + 2) ({(x + 2)}^{3} {(x - 2)}^{2} \sqrt{x + 2})}{(x + 2) \cdot 1}$

Step 5.2.2.2

Cancel the common factor.

$\frac{(x + 2) ({(x + 2)}^{3} {(x - 2)}^{2} \sqrt{x + 2})}{(x + 2) \cdot 1}$

Step 5.2.2.3

Rewrite the expression.

$\frac{{(x + 2)}^{3} {(x - 2)}^{2} \sqrt{x + 2}}{1}$

Step 5.2.2.4

Divide ${(x + 2)}^{3} {(x - 2)}^{2} \sqrt{x + 2}$ by $1$ .

${(x + 2)}^{3} {(x - 2)}^{2} \sqrt{x + 2}$

Step 6

Use the Binomial Theorem.

$(x^{3} + 3 x^{2} \cdot 2 + 3 x \cdot 2^{2} + 2^{3}) {(x - 2)}^{2} \sqrt{x + 2}$

Step 7

Simplify terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $2$ by $3$ .

$(x^{3} + 6 x^{2} + 3 x \cdot 2^{2} + 2^{3}) {(x - 2)}^{2} \sqrt{x + 2}$

Step 7.1.2

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

$(x^{3} + 6 x^{2} + 3 x \cdot 4 + 2^{3}) {(x - 2)}^{2} \sqrt{x + 2}$

Step 7.1.3

Multiply $4$ by $3$ .

$(x^{3} + 6 x^{2} + 12 x + 2^{3}) {(x - 2)}^{2} \sqrt{x + 2}$

Step 7.1.4

Raise $2$ to the power of $3$ .

$(x^{3} + 6 x^{2} + 12 x + 8) {(x - 2)}^{2} \sqrt{x + 2}$

Step 7.2

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$(x^{3} + 6 x^{2} + 12 x + 8) ((x - 2) (x - 2)) \sqrt{x + 2}$

Step 8

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$(x^{3} + 6 x^{2} + 12 x + 8) (x (x - 2) - 2 (x - 2)) \sqrt{x + 2}$

Step 8.2

Apply the distributive property.

$(x^{3} + 6 x^{2} + 12 x + 8) (x \cdot x + x \cdot - 2 - 2 (x - 2)) \sqrt{x + 2}$

Step 8.3

Apply the distributive property.

$(x^{3} + 6 x^{2} + 12 x + 8) (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) \sqrt{x + 2}$

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $x$ by $x$ .

$(x^{3} + 6 x^{2} + 12 x + 8) (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) \sqrt{x + 2}$

Step 9.1.2

Move $- 2$ to the left of $x$ .

$(x^{3} + 6 x^{2} + 12 x + 8) (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) \sqrt{x + 2}$

Step 9.1.3

Multiply $- 2$ by $- 2$ .

$(x^{3} + 6 x^{2} + 12 x + 8) (x^{2} - 2 x - 2 x + 4) \sqrt{x + 2}$

Step 9.2

Subtract $2 x$ from $- 2 x$ .

$(x^{3} + 6 x^{2} + 12 x + 8) (x^{2} - 4 x + 4) \sqrt{x + 2}$

Step 10

Expand $(x^{3} + 6 x^{2} + 12 x + 8) (x^{2} - 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 6 x^{2} x^{2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11

Simplify terms.

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Step 11.1

Simplify each term.

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Step 11.1.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 11.1.1.1

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

$(x^{3 + 2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 6 x^{2} x^{2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.1.2

Add $3$ and $2$ .

$(x^{5} + x^{3} (- 4 x) + x^{3} \cdot 4 + 6 x^{2} x^{2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.2

Rewrite using the commutative property of multiplication.

$(x^{5} - 4 x^{3} x + x^{3} \cdot 4 + 6 x^{2} x^{2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.3

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 11.1.3.1

Move $x$ .

$(x^{5} - 4 (x \cdot x^{3}) + x^{3} \cdot 4 + 6 x^{2} x^{2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.3.2

Multiply $x$ by $x^{3}$ .

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Step 11.1.3.2.1

Raise $x$ to the power of $1$ .

$(x^{5} - 4 (x^{1} x^{3}) + x^{3} \cdot 4 + 6 x^{2} x^{2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.3.2.2

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

$(x^{5} - 4 x^{1 + 3} + x^{3} \cdot 4 + 6 x^{2} x^{2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.3.3

Add $1$ and $3$ .

$(x^{5} - 4 x^{4} + x^{3} \cdot 4 + 6 x^{2} x^{2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.4

Move $4$ to the left of $x^{3}$ .

$(x^{5} - 4 x^{4} + 4 \cdot x^{3} + 6 x^{2} x^{2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.5

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

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Step 11.1.5.1

Move $x^{2}$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 (x^{2} x^{2}) + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.5.2

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

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{2 + 2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.5.3

Add $2$ and $2$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.6

Rewrite using the commutative property of multiplication.

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} + 6 \cdot - 4 x^{2} x + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.7

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

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Step 11.1.7.1

Move $x$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} + 6 \cdot - 4 (x \cdot x^{2}) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.7.2

Multiply $x$ by $x^{2}$ .

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Step 11.1.7.2.1

Raise $x$ to the power of $1$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} + 6 \cdot - 4 (x^{1} x^{2}) + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.7.2.2

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

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} + 6 \cdot - 4 x^{1 + 2} + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.7.3

Add $1$ and $2$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} + 6 \cdot - 4 x^{3} + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.8

Multiply $6$ by $- 4$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 6 x^{2} \cdot 4 + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.9

Multiply $4$ by $6$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 x \cdot x^{2} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.10

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

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Step 11.1.10.1

Move $x^{2}$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 (x^{2} x) + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.10.2

Multiply $x^{2}$ by $x$ .

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Step 11.1.10.2.1

Raise $x$ to the power of $1$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 (x^{2} x^{1}) + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.10.2.2

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

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 x^{2 + 1} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.10.3

Add $2$ and $1$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 x^{3} + 12 x (- 4 x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.11

Rewrite using the commutative property of multiplication.

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 x^{3} + 12 \cdot - 4 x \cdot x + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.12

Multiply $x$ by $x$ by adding the exponents.

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Step 11.1.12.1

Move $x$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 x^{3} + 12 \cdot - 4 (x \cdot x) + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.12.2

Multiply $x$ by $x$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 x^{3} + 12 \cdot - 4 x^{2} + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.13

Multiply $12$ by $- 4$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 x^{3} - 48 x^{2} + 12 x \cdot 4 + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.14

Multiply $4$ by $12$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 x^{3} - 48 x^{2} + 48 x + 8 x^{2} + 8 (- 4 x) + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.15

Multiply $- 4$ by $8$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 x^{3} - 48 x^{2} + 48 x + 8 x^{2} - 32 x + 8 \cdot 4) \sqrt{x + 2}$

Step 11.1.16

Multiply $8$ by $4$ .

$(x^{5} - 4 x^{4} + 4 x^{3} + 6 x^{4} - 24 x^{3} + 24 x^{2} + 12 x^{3} - 48 x^{2} + 48 x + 8 x^{2} - 32 x + 32) \sqrt{x + 2}$

Step 11.2

Simplify terms.

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Step 11.2.1

Add $- 4 x^{4}$ and $6 x^{4}$ .

$(x^{5} + 2 x^{4} + 4 x^{3} - 24 x^{3} + 24 x^{2} + 12 x^{3} - 48 x^{2} + 48 x + 8 x^{2} - 32 x + 32) \sqrt{x + 2}$

Step 11.2.2

Subtract $24 x^{3}$ from $4 x^{3}$ .

$(x^{5} + 2 x^{4} - 20 x^{3} + 24 x^{2} + 12 x^{3} - 48 x^{2} + 48 x + 8 x^{2} - 32 x + 32) \sqrt{x + 2}$

Step 11.2.3

Add $- 20 x^{3}$ and $12 x^{3}$ .

$(x^{5} + 2 x^{4} - 8 x^{3} + 24 x^{2} - 48 x^{2} + 48 x + 8 x^{2} - 32 x + 32) \sqrt{x + 2}$

Step 11.2.4

Subtract $48 x^{2}$ from $24 x^{2}$ .

$(x^{5} + 2 x^{4} - 8 x^{3} - 24 x^{2} + 48 x + 8 x^{2} - 32 x + 32) \sqrt{x + 2}$

Step 11.2.5

Add $- 24 x^{2}$ and $8 x^{2}$ .

$(x^{5} + 2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x - 32 x + 32) \sqrt{x + 2}$

Step 11.2.6

Subtract $32 x$ from $48 x$ .

$(x^{5} + 2 x^{4} - 8 x^{3} - 16 x^{2} + 16 x + 32) \sqrt{x + 2}$

Step 11.2.7

Apply the distributive property.

$x^{5} \sqrt{x + 2} + 2 x^{4} \sqrt{x + 2} - 8 x^{3} \sqrt{x + 2} - 16 x^{2} \sqrt{x + 2} + 16 x \sqrt{x + 2} + 32 \sqrt{x + 2}$