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

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

To divide by a fraction, multiply by its reciprocal.

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

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

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

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

Raise $x$ to the power of $1$ .

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

Factor $x$ out of $x^{1}$ .

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

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Simplify the denominator.

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

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

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 $b = 2$ .

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

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Write the factored form using these integers.

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

Simplify the denominator.

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

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

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 $b = 1$ .

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

Cancel the common factor of $x + 1$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{x}{x - 2}$ by $\frac{x + 3}{x - 1}$ .

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

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

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