Answer
$2t^2(s^3+2t)(s^3+3t)$
Work Step by Step
Factoring the $GCF=
2t^2
$, then the given expression, $
2s^6t^2+10s^3t^3+12t^4
$ is equivalent to
\begin{array}{l}
2t^2(s^6+5s^3t+6t^2)
.\end{array}
The two numbers whose product is $ac=
1(6)=6
$ and whose sum is $b=
5
$ are $\{
2,3
\}$. Using these two numbers to decompose the middle term of the expression, $
2t^2(s^6+5s^3t+6t^2)
,$ then the factored form is
\begin{array}{l}
2t^2(s^6+2s^3t+3s^3t+6t^2)
\\\\=
2t^2[(s^6+2s^3t)+(3s^3t+6t^2)]
\\\\=
2t^2[s^3(s^3+2t)+3t(s^3+2t)]
\\\\=
2t^2[(s^3+2t)(s^3+3t)]
\\\\=
2t^2(s^3+2t)(s^3+3t)
.\end{array}
