Answer
$6a^{8}(a+7)(a-2)$
Work Step by Step
Factoring the $GCF=
6a^8
$, the given expression, $
6a^{10}+30a^9-84a^8
,$ is equivalent to
\begin{array}{l}
6a^{8}(a^2+5a-14)
.\end{array}
The 2 numbers whose product is $ac=
1(-14)=-14
$ and whose sum is $b=
5
$ are $\left\{
7,-2
\right\}.$ Using these two numbers to decompose the middle term of the above trinomial, then
\begin{array}{l}
6a^{8}(a^2+7a-2a-14)
\\\\=
6a^{8}[(a^2+7a)-(2a+14)]
\\\\=
6a^{8}[a(a+7)-2(a+7)]
\\\\=
6a^{8}[(a+7)(a-2)]
\\\\=
6a^{8}(a+7)(a-2)
.\end{array}