Answer
$3x(7x+1)(8x+1)$
Work Step by Step
Factoring the $GCF=
3x
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
168x^3+45x^2+3x
\\\\=
3x(56x^2+15x+1)
.\end{array}
Using the factoring of trinomials in the form $ax^2+bx+c,$ the $\text{
expression
}$
\begin{array}{l}\require{cancel}
3x(56x^2+15x+1)
\end{array} has $ac=
56(1)=56
$ and $b=
15
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
8,7
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
3x(56x^2+8x+7x+1)
.\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}
3x[(56x^2+8x)+(7x+1)]
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
3x[8x(7x+1)+(7x+1)]
.\end{array}
Factoring the $GCF=
(7x+1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
3x[(7x+1)(8x+1)]
\\\\=
3x(7x+1)(8x+1)
.\end{array}