Divide ((x^2-4x-5)/(x^2-3x+2))÷((x^2-3x-10)/(x^2-4))

$\frac{x^{2} - 4 x - 5}{x^{2} - 3 x + 2} \div \frac{x^{2} - 3 x - 10}{x^{2} - 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 4 x - 5}{x^{2} - 3 x + 2} \cdot \frac{x^{2} - 4}{x^{2} - 3 x - 10}$

Step 2

Factor $x^{2} - 4 x - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Write the factored form using these integers.

$\frac{(x - 5) (x + 1)}{x^{2} - 3 x + 2} \cdot \frac{x^{2} - 4}{x^{2} - 3 x - 10}$

Step 3

Factor $x^{2} - 3 x + 2$ 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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

$\frac{(x - 5) (x + 1)}{(x - 2) (x - 1)} \cdot \frac{x^{2} - 4}{x^{2} - 3 x - 10}$

Step 4

Simplify the numerator.

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

$\frac{(x - 5) (x + 1)}{(x - 2) (x - 1)} \cdot \frac{x^{2} - 2^{2}}{x^{2} - 3 x - 10}$

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 $b = 2$ .

$\frac{(x - 5) (x + 1)}{(x - 2) (x - 1)} \cdot \frac{(x + 2) (x - 2)}{x^{2} - 3 x - 10}$

Step 5

Factor $x^{2} - 3 x - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

$\frac{(x - 5) (x + 1)}{(x - 2) (x - 1)} \cdot \frac{(x + 2) (x - 2)}{(x - 5) (x + 2)}$

Step 6

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{(x - 5) (x + 1)}{(x - 2) (x - 1)} \cdot \frac{(x + 2) (x - 2)}{(x - 5) (x + 2)}$

Rewrite the expression.

$\frac{x + 1}{(x - 2) (x - 1)} \cdot \frac{(x + 2) (x - 2)}{x + 2}$

Step 7

Cancel the common factor of $x - 2$ .

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

$\frac{x + 1}{(x - 2) (x - 1)} \cdot \frac{(x - 2) (x + 2)}{x + 2}$

Cancel the common factor.

$\frac{x + 1}{(x - 2) (x - 1)} \cdot \frac{(x - 2) (x + 2)}{x + 2}$

Rewrite the expression.

$\frac{x + 1}{x - 1} \cdot \frac{x + 2}{x + 2}$

Step 8

Multiply $\frac{x + 1}{x - 1}$ by $\frac{x + 2}{x + 2}$ .

$\frac{(x + 1) (x + 2)}{(x - 1) (x + 2)}$

Step 9

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{(x + 1) (x + 2)}{(x - 1) (x + 2)}$

Rewrite the expression.

$\frac{x + 1}{x - 1}$

Divide ((x^2-4x-5)/(x^2-3x+2))÷((x^2-3x-10)/(x^2-4))

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