Answer
$2(s+t)(4s+7t)$
Work Step by Step
Factoring the $GCF=
2
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
8s^2+22st+14t^2
\\\\=
2(4s^2+11st+7t^2)
.\end{array}
Using the factoring of trinomials in the form $ax^2+bx+c,$ the $\text{
expression
}$
\begin{array}{l}\require{cancel}
2(4s^2+11st+7t^2)
\end{array} has $ac=
4(7)=28
$ and $b=
11
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
4,7
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
2(4s^2+4st+7st+7t^2)
.\end{array}
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
2[(4s^2+4st)+(7st+7t^2)]
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
2[4s(s+t)+7t(s+t)]
.\end{array}
Factoring the $GCF=
(s+t)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
2[(s+t)(4s+7t)]
\\\\=
2(s+t)(4s+7t)
.\end{array}