Simplify ((m^2-m-12)/(8m+80))÷((-m^2-8m+48)/(11m+110))

$\frac{m^{2} - m - 12}{8 m + 80} \div \frac{- m^{2} - 8 m + 48}{11 m + 110}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{m^{2} - m - 12}{8 m + 80} \cdot \frac{11 m + 110}{- m^{2} - 8 m + 48}$

Step 2

Factor $m^{2} - m - 12$ 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 $- 1$ .

$- 4, 3$

Write the factored form using these integers.

$\frac{(m - 4) (m + 3)}{8 m + 80} \cdot \frac{11 m + 110}{- m^{2} - 8 m + 48}$

Step 3

Simplify with factoring out.

Factor $8$ out of $8 m + 80$ .

Factor $8$ out of $8 m$ .

$\frac{(m - 4) (m + 3)}{8 (m) + 80} \cdot \frac{11 m + 110}{- m^{2} - 8 m + 48}$

Factor $8$ out of $80$ .

$\frac{(m - 4) (m + 3)}{8 m + 8 \cdot 10} \cdot \frac{11 m + 110}{- m^{2} - 8 m + 48}$

Factor $8$ out of $8 m + 8 \cdot 10$ .

$\frac{(m - 4) (m + 3)}{8 (m + 10)} \cdot \frac{11 m + 110}{- m^{2} - 8 m + 48}$

Factor $11$ out of $11 m + 110$ .

Factor $11$ out of $11 m$ .

$\frac{(m - 4) (m + 3)}{8 (m + 10)} \cdot \frac{11 (m) + 110}{- m^{2} - 8 m + 48}$

Factor $11$ out of $110$ .

$\frac{(m - 4) (m + 3)}{8 (m + 10)} \cdot \frac{11 m + 11 \cdot 10}{- m^{2} - 8 m + 48}$

Factor $11$ out of $11 m + 11 \cdot 10$ .

$\frac{(m - 4) (m + 3)}{8 (m + 10)} \cdot \frac{11 (m + 10)}{- m^{2} - 8 m + 48}$

Step 4

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 $a \cdot c = - 1 \cdot 48 = - 48$ and whose sum is $b = - 8$ .

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

$\frac{(m - 4) (m + 3)}{8 (m + 10)} \cdot \frac{11 (m + 10)}{- m^{2} - 8 (m) + 48}$

Rewrite $- 8$ as $4$ plus $- 12$

$\frac{(m - 4) (m + 3)}{8 (m + 10)} \cdot \frac{11 (m + 10)}{- m^{2} + (4 - 12) m + 48}$

Apply the distributive property.

$\frac{(m - 4) (m + 3)}{8 (m + 10)} \cdot \frac{11 (m + 10)}{- m^{2} + 4 m - 12 m + 48}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(m - 4) (m + 3)}{8 (m + 10)} \cdot \frac{11 (m + 10)}{(- m^{2} + 4 m) - 12 m + 48}$

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

$\frac{(m - 4) (m + 3)}{8 (m + 10)} \cdot \frac{11 (m + 10)}{m (- m + 4) + 12 (- m + 4)}$

Factor the polynomial by factoring out the greatest common factor, $- m + 4$ .

$\frac{(m - 4) (m + 3)}{8 (m + 10)} \cdot \frac{11 (m + 10)}{(- m + 4) (m + 12)}$

Step 5

Cancel the common factor of $m + 10$ .

Factor $m + 10$ out of $8 (m + 10)$ .

$\frac{(m - 4) (m + 3)}{(m + 10) \cdot 8} \cdot \frac{11 (m + 10)}{(- m + 4) (m + 12)}$

Factor $m + 10$ out of $11 (m + 10)$ .

$\frac{(m - 4) (m + 3)}{(m + 10) \cdot 8} \cdot \frac{(m + 10) \cdot 11}{(- m + 4) (m + 12)}$

Cancel the common factor.

$\frac{(m - 4) (m + 3)}{(m + 10) \cdot 8} \cdot \frac{(m + 10) \cdot 11}{(- m + 4) (m + 12)}$

Rewrite the expression.

$\frac{(m - 4) (m + 3)}{8} \cdot \frac{11}{(- m + 4) (m + 12)}$

Step 6

Multiply $\frac{(m - 4) (m + 3)}{8}$ by $\frac{11}{(- m + 4) (m + 12)}$ .

$\frac{(m - 4) (m + 3) \cdot 11}{8 ((- m + 4) (m + 12))}$

Step 7

Cancel the common factor of $m - 4$ and $- m + 4$ .

Factor $- 1$ out of $m$ .

$\frac{(- 1 (- m) - 4) (m + 3) \cdot 11}{8 ((- m + 4) (m + 12))}$

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{(- 1 (- m) - 1 (4)) (m + 3) \cdot 11}{8 ((- m + 4) (m + 12))}$

Factor $- 1$ out of $- 1 (- m) - 1 (4)$ .

$\frac{- 1 (- m + 4) (m + 3) \cdot 11}{8 ((- m + 4) (m + 12))}$

Cancel the common factor.

$\frac{- 1 (- m + 4) (m + 3) \cdot 11}{8 (- m + 4) (m + 12)}$

Rewrite the expression.

$\frac{(- 1 (m + 3)) \cdot 11}{8 (m + 12)}$

Step 8

Simplify the expression.

Multiply $11$ by $- 1$ .

$\frac{- 11 (m + 3)}{8 (m + 12)}$

Move the negative in front of the fraction.

$- \frac{11 (m + 3)}{8 (m + 12)}$

Simplify ((m^2-m-12)/(8m+80))÷((-m^2-8m+48)/(11m+110))

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