Simplify ((x^2-9x+20)/(x-5))÷((x^2-16)/(x^2+4x))

$\frac{x^{2} - 9 x + 20}{x - 5} \div \frac{x^{2} - 16}{x^{2} + 4 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 9 x + 20}{x - 5} \cdot \frac{x^{2} + 4 x}{x^{2} - 16}$

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 $- 9$ .

$- 5, - 4$

Write the factored form using these integers.

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

Step 3

Simplify terms.

Cancel the common factor of $x - 5$ .

Cancel the common factor.

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

Divide $x - 4$ by $1$ .

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

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

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

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

Factor $x$ out of $4 x$ .

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

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

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

Step 4

Simplify the denominator.

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

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

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 $b = 4$ .

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

Step 5

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x - 4$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

$x$

Simplify ((x^2-9x+20)/(x-5))÷((x^2-16)/(x^2+4x))

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