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

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

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 $- 5$ .

$- 6, 1$

Write the factored form using these integers.

$\frac{(x - 6) (x + 1)}{2 x + 6} \cdot \frac{4 x + 12}{x^{2} - 3 x - 4}$

Step 3

Factor $2$ out of $2 x + 6$ .

Factor $2$ out of $2 x$ .

$\frac{(x - 6) (x + 1)}{2 (x) + 6} \cdot \frac{4 x + 12}{x^{2} - 3 x - 4}$

Factor $2$ out of $6$ .

$\frac{(x - 6) (x + 1)}{2 x + 2 \cdot 3} \cdot \frac{4 x + 12}{x^{2} - 3 x - 4}$

Factor $2$ out of $2 x + 2 \cdot 3$ .

$\frac{(x - 6) (x + 1)}{2 (x + 3)} \cdot \frac{4 x + 12}{x^{2} - 3 x - 4}$

Step 4

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

Factor $4$ out of $4 x$ .

$\frac{(x - 6) (x + 1)}{2 (x + 3)} \cdot \frac{4 (x) + 12}{x^{2} - 3 x - 4}$

Factor $4$ out of $12$ .

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

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

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

Step 5

Factor $x^{2} - 3 x - 4$ 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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Write the factored form using these integers.

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

Step 6

Cancel the common factor of $x + 1$ .

Factor $x + 1$ out of $(x - 6) (x + 1)$ .

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

Factor $x + 1$ out of $(x - 4) (x + 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

Cancel the common factor of $2 (x + 3)$ .

Factor $2 (x + 3)$ out of $4 (x + 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$(x - 6) \frac{2}{x - 4}$

Step 8

Multiply $x - 6$ by $\frac{2}{x - 4}$ .

$\frac{(x - 6) \cdot 2}{x - 4}$

Step 9

Move $2$ to the left of $x - 6$ .

$\frac{2 (x - 6)}{x - 4}$

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

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