Answer
$t(t+2)(7t+1)$
Work Step by Step
Factoring the $GCF=
t
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
7t^3+15t^2+2t
\\\\=
t(7t^2+15t+2)
.\end{array}
Using the factoring of trinomials in the form $ax^2+bx+c,$ the $\text{
expression
}$
\begin{array}{l}\require{cancel}
t(7t^2+15t+2)
\end{array} has $ac=
7(2)=14
$ and $b=
15
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
14,1
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
t(7t^2+14t+1t+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}
t[(7t^2+14t)+(1t+2)]
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
t[7t(t+2)+(t+2)]
.\end{array}
Factoring the $GCF=
(t+2)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
t[(t+2)(7t+1)]
\\\\=
t(t+2)(7t+1)
.\end{array}