Answer
$-a^4(a+15)(a-6)$
Work Step by Step
Factoring the negative $GCF=
-a^4
$, the given expression, $
-a^6-9a^5+90a^4
,$ is equivalent to
\begin{array}{l}
-a^4(a^2+9a-90)
.\end{array}
The 2 numbers whose product is $ac=
1(-90)=-90
$ and whose sum is $b=
9
$ are $\left\{
15,-6
\right\}.$ Using these two numbers to decompose the middle term of the above trinomial, then
\begin{array}{l}
-a^4(a^2+15a-6a-90)
\\\\=
-a^4[(a^2+15a)-(6a+90)]
\\\\=
-a^4[a(a+15)-6(a+15)]
\\\\=
-a^4[(a+15)(a-6)]
\\\\=
-a^4(a+15)(a-6)
.\end{array}
