Simplify ((x^2-4)/(4x+12))÷((x^2+5x+6)/(x^2+3x))

$\frac{x^{2} - 4}{4 x + 12} \div \frac{x^{2} + 5 x + 6}{x^{2} + 3 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 4}{4 x + 12} \cdot \frac{x^{2} + 3 x}{x^{2} + 5 x + 6}$

Step 2

Simplify the numerator.

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

$\frac{x^{2} - 2^{2}}{4 x + 12} \cdot \frac{x^{2} + 3 x}{x^{2} + 5 x + 6}$

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 $b = 2$ .

$\frac{(x + 2) (x - 2)}{4 x + 12} \cdot \frac{x^{2} + 3 x}{x^{2} + 5 x + 6}$

Step 3

Simplify with factoring out.

Factor $4$ out of $4 x + 12$ .

Factor $4$ out of $4 x$ .

$\frac{(x + 2) (x - 2)}{4 (x) + 12} \cdot \frac{x^{2} + 3 x}{x^{2} + 5 x + 6}$

Factor $4$ out of $12$ .

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

Factor $4$ out of $4 x + 4 \cdot 3$ .

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

Factor $x$ out of $x^{2} + 3 x$ .

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

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

Factor $x$ out of $3 x$ .

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

Factor $x$ out of $x \cdot x + x \cdot 3$ .

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

Step 4

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.

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

Step 5

Simplify terms.

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 3$ .

Factor $x + 3$ out of $4 (x + 3)$ .

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

Factor $x + 3$ out of $x (x + 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{x - 2}{4}$ by $\frac{x}{x + 3}$ .

$\frac{(x - 2) x}{4 (x + 3)}$

Reorder factors in $\frac{(x - 2) x}{4 (x + 3)}$ .

$\frac{x (x - 2)}{4 (x + 3)}$

Simplify ((x^2-4)/(4x+12))÷((x^2+5x+6)/(x^2+3x))

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