Answer
$\frac{2(m + 1)}{m + 4}$
Work Step by Step
Let's factor the denominators first:
First denominator:
$m^2 - m - 20 = (m - 5)(m + 4)$
Second denominator:
$m^2 - 4m - 5 = (m - 5)(m + 1)$
Now, let's put the factored denominators back into the expression:
$\frac{2m + 4}{(m - 5)(m + 4)} \div \frac{m + 2}{(m - 5)(m + 1)}$
To divide two expressions, we multiply the first one with the reciprocal of the second one:
$\frac{2m + 4}{(m - 5)(m + 4)} • \frac{(m - 5)(m + 1)}{m + 2}$
Combine the two rational expressions into one by multiplying numerators together and denominators together:
$\frac{(2m + 4)(m - 5)(m + 1)}{(m - 5)(m + 4)(m + 2)}$
Now, we want to cancel out the factors that are common in the numerators and denominators of each rational expression:
$\frac{(2m + 4)(m + 1)}{(m + 4)(m + 2)}$
Factor out the common term in the numerator:
$\frac{2(m + 2)(m + 1)}{(m + 4)(m + 2)}$
Finally, we can cancel out the common factor in both the numerator and denominator:
$\frac{2(m + 1)}{m + 4}$
