Simplify ((8x+32)/(x^2+8x+16))÷((x^2-6x)/(x^2-2x-24))

$\frac{8 x + 32}{x^{2} + 8 x + 16} \div \frac{x^{2} - 6 x}{x^{2} - 2 x - 24}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{8 x + 32}{x^{2} + 8 x + 16} \cdot \frac{x^{2} - 2 x - 24}{x^{2} - 6 x}$

Step 2

Factor $8$ out of $8 x + 32$ .

Factor $8$ out of $8 x$ .

$\frac{8 (x) + 32}{x^{2} + 8 x + 16} \cdot \frac{x^{2} - 2 x - 24}{x^{2} - 6 x}$

Factor $8$ out of $32$ .

$\frac{8 x + 8 \cdot 4}{x^{2} + 8 x + 16} \cdot \frac{x^{2} - 2 x - 24}{x^{2} - 6 x}$

Factor $8$ out of $8 x + 8 \cdot 4$ .

$\frac{8 (x + 4)}{x^{2} + 8 x + 16} \cdot \frac{x^{2} - 2 x - 24}{x^{2} - 6 x}$

Step 3

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$\frac{8 (x + 4)}{x^{2} + 8 x + 4^{2}} \cdot \frac{x^{2} - 2 x - 24}{x^{2} - 6 x}$

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

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

$\frac{8 (x + 4)}{x^{2} + 2 \cdot x \cdot 4 + 4^{2}} \cdot \frac{x^{2} - 2 x - 24}{x^{2} - 6 x}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

$\frac{8 (x + 4)}{{(x + 4)}^{2}} \cdot \frac{x^{2} - 2 x - 24}{x^{2} - 6 x}$

Step 4

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

Factor $x + 4$ out of $8 (x + 4)$ .

$\frac{(x + 4) \cdot 8}{{(x + 4)}^{2}} \cdot \frac{x^{2} - 2 x - 24}{x^{2} - 6 x}$

Cancel the common factors.

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

$\frac{(x + 4) \cdot 8}{(x + 4) (x + 4)} \cdot \frac{x^{2} - 2 x - 24}{x^{2} - 6 x}$

Cancel the common factor.

$\frac{(x + 4) \cdot 8}{(x + 4) (x + 4)} \cdot \frac{x^{2} - 2 x - 24}{x^{2} - 6 x}$

Rewrite the expression.

$\frac{8}{x + 4} \cdot \frac{x^{2} - 2 x - 24}{x^{2} - 6 x}$

Step 5

Factor $x^{2} - 2 x - 24$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Write the factored form using these integers.

$\frac{8}{x + 4} \cdot \frac{(x - 6) (x + 4)}{x^{2} - 6 x}$

Step 6

Simplify terms.

Factor $x$ out of $x^{2} - 6 x$ .

Factor $x$ out of $x^{2}$ .

$\frac{8}{x + 4} \cdot \frac{(x - 6) (x + 4)}{x \cdot x - 6 x}$

Factor $x$ out of $- 6 x$ .

$\frac{8}{x + 4} \cdot \frac{(x - 6) (x + 4)}{x \cdot x + x \cdot - 6}$

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

$\frac{8}{x + 4} \cdot \frac{(x - 6) (x + 4)}{x (x - 6)}$

Cancel the common factor of $x + 4$ .

Factor $x + 4$ out of $(x - 6) (x + 4)$ .

$\frac{8}{x + 4} \cdot \frac{(x + 4) (x - 6)}{x (x - 6)}$

Cancel the common factor.

$\frac{8}{x + 4} \cdot \frac{(x + 4) (x - 6)}{x (x - 6)}$

Rewrite the expression.

$8 \frac{x - 6}{x (x - 6)}$

Combine $8$ and $\frac{x - 6}{x (x - 6)}$ .

$\frac{8 (x - 6)}{x (x - 6)}$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{8 (x - 6)}{x (x - 6)}$

Rewrite the expression.

$\frac{8}{x}$

Simplify ((8x+32)/(x^2+8x+16))÷((x^2-6x)/(x^2-2x-24))

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