Answer
$a^3(a-5b)(a+b)$
Work Step by Step
Factoring the $GCF=
a^3
$, then the given expression, $
a^5-4a^4b-5a^3b^2
$ is equivalent to
\begin{array}{l}
a^3(a^2-4ab-5b^2)
.\end{array}
The two numbers whose product is $ac=
1(-5)=-5
$ and whose sum is $b=
-4
$ are $\{
-5,1
\}$. Using these two numbers to decompose the middle term of the expression, $
a^3(a^2-4ab-5b^2)
,$ then the factored form is
\begin{array}{l}
a^3(a^2-5ab+ab-5b^2)
\\\\=
a^3[(a^2-5ab)+(ab-5b^2)]
\\\\=
a^3[a(a-5b)+b(a-5b)]
\\\\=
a^3[(a-5b)(a+b)]
\\\\=
a^3(a-5b)(a+b)
.\end{array}
