# How do you simplify ( 5a^2 + 20a) /( a^3-2a^2) * ( a^2-a-12) / (a^2-16)?

Jul 26, 2015

$a \frac{5 a + 20}{a} ^ 2 \left(a - 2\right)$ . $\left(a - 4\right) \frac{a + 3}{a - 4} ^ 2$

#### Explanation:

simplyfing the first equation:

having a common factor "a"
a(5a+20)

simplifying The denominator:
having a common factor " ${a}^{2}$ "
${a}^{2}$ (a-2)

Moving to the second equation:

The numerator:
${a}^{2}$-a- 12
This equation cannot be solved by the common factor method, because -12 has no "a".
However, It can be solved by another method:
opening 2 different parenthesis
(a-4).(a+3)

The dominator:
having the power common factor
${\left(a - 4\right)}^{2}$

Jul 26, 2015

By factoring each expression in the numerator(top) and the denominator(bottom) and then cancelling out the commons.

#### Explanation:

There are $4$ expressions. First, each expression must be factored.

Here's how we do it:

$\textcolor{red}{\left(1\right)} 5 {a}^{2} + 20 a = a \left(5 a + 20\right) = 5 a \left(a + 4\right)$

$\textcolor{red}{\left(2\right)} {a}^{3} - 2 {a}^{2} = {a}^{2} \left(a - 2\right)$

$\textcolor{red}{\left(3\right)} {a}^{2} - a - 12 = {a}^{2} - 4 a + 3 a - 12 = a \left(a - 4\right) + 3 \left(a - 4\right) = \left(a + 3\right) \left(a - 4\right)$

$\textcolor{red}{\left(4\right)} {a}^{2} - 16 = {a}^{2} - {4}^{2}$

This is an expression of the form : $\left(A + B\right) \left(A - B\right) = {A}^{2} - {B}^{2}$

Hence,$\textcolor{red}{\left(4\right)} {a}^{2} - 16 = \left(a - 4\right) \left(a + 4\right)$

$\implies \frac{5 {a}^{2} + 20 a}{{a}^{3} - 2 {a}^{2}} \cdot \frac{{a}^{2} - a - 20}{{a}^{2} - 16} \text{ }$ becomes

(5acolor(red)cancel(color(black)((a+4))))/(a^2(a-2))*(color(green)cancel(color(black)((a-4)))(a+3))/(color(green)cancel(color(black)((a-4))) color(red)cancel(color(black)((a+4))))=(5a(a+3))/(a^2(a-2))=color(blue)((5(a+3))/(a(a-2)))