Divide ((x^2+9x+20)/(x^2-25))÷((x+4)/(x-4))

$\frac{x^{2} + 9 x + 20}{x^{2} - 25} \div \frac{x + 4}{x - 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 9 x + 20}{x^{2} - 25} \cdot \frac{x - 4}{x + 4}$

Step 2

Factor $x^{2} + 9 x + 20$ 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 $20$ and whose sum is $9$ .

$4, 5$

Write the factored form using these integers.

$\frac{(x + 4) (x + 5)}{x^{2} - 25} \cdot \frac{x - 4}{x + 4}$

Step 3

Simplify the denominator.

Rewrite $25$ as $5^{2}$ .

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

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 $b = 5$ .

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

Step 4

Cancel the common factor of $x + 4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Cancel the common factor of $x + 5$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{x - 5} (x - 4)$

Step 6

Multiply $\frac{1}{x - 5}$ by $x - 4$ .

$\frac{x - 4}{x - 5}$

Divide ((x^2+9x+20)/(x^2-25))÷((x+4)/(x-4))

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