Simplify ((x^2-x-2)/(x^2+6x+5))÷((2x^2-3x-2)/(4x^2-1))

$\frac{x^{2} - x - 2}{x^{2} + 6 x + 5} \div \frac{2 x^{2} - 3 x - 2}{4 x^{2} - 1}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - x - 2}{x^{2} + 6 x + 5} \cdot \frac{4 x^{2} - 1}{2 x^{2} - 3 x - 2}$

Factor $x^{2} - x - 2$ 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 $- 1$ .

$- 2, 1$

Write the factored form using these integers.

$\frac{(x - 2) (x + 1)}{x^{2} + 6 x + 5} \cdot \frac{4 x^{2} - 1}{2 x^{2} - 3 x - 2}$

Factor $x^{2} + 6 x + 5$ 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 $6$ .

$1, 5$

Write the factored form using these integers.

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

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Rewrite $1$ as $1^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

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

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 x$ .

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

Rewrite $- 3$ as $1$ plus $- 4$

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

Apply the distributive property.

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

Multiply $x$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

Cancel the common factor of $x - 2$ .

Factor $x - 2$ out of $(2 x + 1) (x - 2)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{1}{x + 5}$ by $\frac{(2 x + 1) (2 x - 1)}{2 x + 1}$ .

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

Cancel the common factor of $2 x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 x - 1}{x + 5}$

Simplify ((x^2-x-2)/(x^2+6x+5))÷((2x^2-3x-2)/(4x^2-1))

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