Simplify ((m^2-2m-8)/(8m+24))÷((2m-8)/(m^2+7m+12))

$\frac{m^{2} - 2 m - 8}{8 m + 24} \div \frac{2 m - 8}{m^{2} + 7 m + 12}$

To divide by a fraction, multiply by its reciprocal.

$\frac{m^{2} - 2 m - 8}{8 m + 24} \cdot \frac{m^{2} + 7 m + 12}{2 m - 8}$

Factor $m^{2} - 2 m - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Write the factored form using these integers.

$\frac{(m - 4) (m + 2)}{8 m + 24} \cdot \frac{m^{2} + 7 m + 12}{2 m - 8}$

Factor $8$ out of $8 m + 24$ .

Factor $8$ out of $8 m$ .

$\frac{(m - 4) (m + 2)}{8 (m) + 24} \cdot \frac{m^{2} + 7 m + 12}{2 m - 8}$

Factor $8$ out of $24$ .

$\frac{(m - 4) (m + 2)}{8 m + 8 \cdot 3} \cdot \frac{m^{2} + 7 m + 12}{2 m - 8}$

Factor $8$ out of $8 m + 8 \cdot 3$ .

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{m^{2} + 7 m + 12}{2 m - 8}$

Factor $m^{2} + 7 m + 12$ 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 $12$ and whose sum is $7$ .

$3, 4$

Write the factored form using these integers.

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{2 m - 8}$

Simplify terms.

Factor $2$ out of $2 m - 8$ .

Factor $2$ out of $2 m$ .

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{2 (m) - 8}$

Factor $2$ out of $- 8$ .

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{2 m + 2 \cdot - 4}$

Factor $2$ out of $2 m + 2 \cdot - 4$ .

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{2 (m - 4)}$

Cancel the common factor of $m - 4$ .

Factor $m - 4$ out of $2 (m - 4)$ .

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{(m - 4) \cdot 2}$

Cancel the common factor.

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{(m - 4) \cdot 2}$

Rewrite the expression.

$\frac{m + 2}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{2}$

Cancel the common factor of $m + 3$ .

Factor $m + 3$ out of $8 (m + 3)$ .

$\frac{m + 2}{(m + 3) \cdot 8} \cdot \frac{(m + 3) (m + 4)}{2}$

Cancel the common factor.

$\frac{m + 2}{(m + 3) \cdot 8} \cdot \frac{(m + 3) (m + 4)}{2}$

Rewrite the expression.

$\frac{m + 2}{8} \cdot \frac{m + 4}{2}$

Multiply $\frac{m + 2}{8}$ by $\frac{m + 4}{2}$ .

$\frac{(m + 2) (m + 4)}{8 \cdot 2}$

Multiply $8$ by $2$ .

$\frac{(m + 2) (m + 4)}{16}$

Simplify ((m^2-2m-8)/(8m+24))÷((2m-8)/(m^2+7m+12))

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