Divide ((x^2+5x-36)/(x+5))÷((x^2-16)/(x+4))

$\frac{x^{2} + 5 x - 36}{x + 5} \div \frac{x^{2} - 16}{x + 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 5 x - 36}{x + 5} \cdot \frac{x + 4}{x^{2} - 16}$

Step 2

Factor $x^{2} + 5 x - 36$ 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 $- 36$ and whose sum is $5$ .

$- 4, 9$

Write the factored form using these integers.

$\frac{(x - 4) (x + 9)}{x + 5} \cdot \frac{x + 4}{x^{2} - 16}$

Step 3

Simplify the denominator.

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

$\frac{(x - 4) (x + 9)}{x + 5} \cdot \frac{x + 4}{x^{2} - 4^{2}}$

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 $b = 4$ .

$\frac{(x - 4) (x + 9)}{x + 5} \cdot \frac{x + 4}{(x + 4) (x - 4)}$

Step 4

Cancel the common factor of $x - 4$ .

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

$\frac{(x - 4) (x + 9)}{x + 5} \cdot \frac{x + 4}{(x - 4) (x + 4)}$

Cancel the common factor.

$\frac{(x - 4) (x + 9)}{x + 5} \cdot \frac{x + 4}{(x - 4) (x + 4)}$

Rewrite the expression.

$\frac{x + 9}{x + 5} \cdot \frac{x + 4}{x + 4}$

Step 5

Multiply $\frac{x + 9}{x + 5}$ by $\frac{x + 4}{x + 4}$ .

$\frac{(x + 9) (x + 4)}{(x + 5) (x + 4)}$

Step 6

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{(x + 9) (x + 4)}{(x + 5) (x + 4)}$

Rewrite the expression.

$\frac{x + 9}{x + 5}$

Divide ((x^2+5x-36)/(x+5))÷((x^2-16)/(x+4))

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