Divide ((x^2-1)/(3x^2+5x+2))÷((x^2+6x-7)/(x^2-4x-5))

$\frac{x^{2} - 1}{3 x^{2} + 5 x + 2} \div \frac{x^{2} + 6 x - 7}{x^{2} - 4 x - 5}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 1}{3 x^{2} + 5 x + 2} \cdot \frac{x^{2} - 4 x - 5}{x^{2} + 6 x - 7}$

Simplify the numerator.

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

$\frac{x^{2} - 1^{2}}{3 x^{2} + 5 x + 2} \cdot \frac{x^{2} - 4 x - 5}{x^{2} + 6 x - 7}$

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 $b = 1$ .

$\frac{(x + 1) (x - 1)}{3 x^{2} + 5 x + 2} \cdot \frac{x^{2} - 4 x - 5}{x^{2} + 6 x - 7}$

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 $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

$\frac{(x + 1) (x - 1)}{3 x^{2} + 5 (x) + 2} \cdot \frac{x^{2} - 4 x - 5}{x^{2} + 6 x - 7}$

Rewrite $5$ as $2$ plus $3$

$\frac{(x + 1) (x - 1)}{3 x^{2} + (2 + 3) x + 2} \cdot \frac{x^{2} - 4 x - 5}{x^{2} + 6 x - 7}$

Apply the distributive property.

$\frac{(x + 1) (x - 1)}{3 x^{2} + 2 x + 3 x + 2} \cdot \frac{x^{2} - 4 x - 5}{x^{2} + 6 x - 7}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(x + 1) (x - 1)}{(3 x^{2} + 2 x) + 3 x + 2} \cdot \frac{x^{2} - 4 x - 5}{x^{2} + 6 x - 7}$

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

$\frac{(x + 1) (x - 1)}{x (3 x + 2) + 1 (3 x + 2)} \cdot \frac{x^{2} - 4 x - 5}{x^{2} + 6 x - 7}$

Factor the polynomial by factoring out the greatest common factor, $3 x + 2$ .

$\frac{(x + 1) (x - 1)}{(3 x + 2) (x + 1)} \cdot \frac{x^{2} - 4 x - 5}{x^{2} + 6 x - 7}$

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{(x + 1) (x - 1)}{(3 x + 2) (x + 1)} \cdot \frac{x^{2} - 4 x - 5}{x^{2} + 6 x - 7}$

Rewrite the expression.

$\frac{x - 1}{3 x + 2} \cdot \frac{x^{2} - 4 x - 5}{x^{2} + 6 x - 7}$

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.

$\frac{x - 1}{3 x + 2} \cdot \frac{(x - 5) (x + 1)}{x^{2} + 6 x - 7}$

Factor $x^{2} + 6 x - 7$ 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 $- 7$ and whose sum is $6$ .

$- 1, 7$

Write the factored form using these integers.

$\frac{x - 1}{3 x + 2} \cdot \frac{(x - 5) (x + 1)}{(x - 1) (x + 7)}$

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{x - 1}{3 x + 2} \cdot \frac{(x - 5) (x + 1)}{(x - 1) (x + 7)}$

Rewrite the expression.

$\frac{1}{3 x + 2} \cdot \frac{(x - 5) (x + 1)}{x + 7}$

Multiply $\frac{1}{3 x + 2}$ by $\frac{(x - 5) (x + 1)}{x + 7}$ .

$\frac{(x - 5) (x + 1)}{(3 x + 2) (x + 7)}$

Divide ((x^2-1)/(3x^2+5x+2))÷((x^2+6x-7)/(x^2-4x-5))

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