Divide ((x^2-16)/(x+5))÷((x^2-8x+16)/(2x+10))

$\frac{x^{2} - 16}{x + 5} \div \frac{x^{2} - 8 x + 16}{2 x + 10}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 16}{x + 5} \cdot \frac{2 x + 10}{x^{2} - 8 x + 16}$

Step 2

Simplify the numerator.

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

$\frac{x^{2} - 4^{2}}{x + 5} \cdot \frac{2 x + 10}{x^{2} - 8 x + 16}$

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 $b = 4$ .

$\frac{(x + 4) (x - 4)}{x + 5} \cdot \frac{2 x + 10}{x^{2} - 8 x + 16}$

Step 3

Factor $2$ out of $2 x + 10$ .

Factor $2$ out of $2 x$ .

$\frac{(x + 4) (x - 4)}{x + 5} \cdot \frac{2 (x) + 10}{x^{2} - 8 x + 16}$

Factor $2$ out of $10$ .

$\frac{(x + 4) (x - 4)}{x + 5} \cdot \frac{2 x + 2 \cdot 5}{x^{2} - 8 x + 16}$

Factor $2$ out of $2 x + 2 \cdot 5$ .

$\frac{(x + 4) (x - 4)}{x + 5} \cdot \frac{2 (x + 5)}{x^{2} - 8 x + 16}$

Step 4

Factor using the perfect square rule.

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

$\frac{(x + 4) (x - 4)}{x + 5} \cdot \frac{2 (x + 5)}{x^{2} - 8 x + 4^{2}}$

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.

$\frac{(x + 4) (x - 4)}{x + 5} \cdot \frac{2 (x + 5)}{x^{2} - 2 \cdot x \cdot 4 + 4^{2}}$

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

$\frac{(x + 4) (x - 4)}{x + 5} \cdot \frac{2 (x + 5)}{{(x - 4)}^{2}}$

Step 5

Cancel the common factor of $x - 4$ .

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

$\frac{(x - 4) (x + 4)}{x + 5} \cdot \frac{2 (x + 5)}{{(x - 4)}^{2}}$

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

$\frac{(x - 4) (x + 4)}{x + 5} \cdot \frac{2 (x + 5)}{(x - 4) (x - 4)}$

Cancel the common factor.

$\frac{(x - 4) (x + 4)}{x + 5} \cdot \frac{2 (x + 5)}{(x - 4) (x - 4)}$

Rewrite the expression.

$\frac{x + 4}{x + 5} \cdot \frac{2 (x + 5)}{x - 4}$

Step 6

Cancel the common factor of $x + 5$ .

Factor $x + 5$ out of $2 (x + 5)$ .

$\frac{x + 4}{x + 5} \cdot \frac{(x + 5) \cdot 2}{x - 4}$

Cancel the common factor.

$\frac{x + 4}{x + 5} \cdot \frac{(x + 5) \cdot 2}{x - 4}$

Rewrite the expression.

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

Step 7

Multiply $x + 4$ by $\frac{2}{x - 4}$ .

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

Step 8

Move $2$ to the left of $x + 4$ .

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

Divide ((x^2-16)/(x+5))÷((x^2-8x+16)/(2x+10))

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