Answer
$(3x+14)(2x+9)$
Work Step by Step
Let $z=
x+5
.$ The given expression, $
6(x+5)^2-5(x+5)+1
,$ is equivalent to
\begin{align*}
6z^2-5z+1
.\end{align*}
Using the factoring of trinomials in the form $ax^2+bx+c,$ the expression above has $ac=
6(1)=6
$ and $b=
-5
.$
The two numbers with a product of $ac$ and a sum of $b$ are $\left\{
-2,-3
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{align*}
6z^2-2z-3z+1
.\end{align*}
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{align*}
(6z^2-2z)-(3z-1)
.\end{align*}
Factoring the $GCF$ in each group results to
\begin{align*}
2z(3z-1)-(3z-1)
.\end{align*}
Factoring the $GCF=
(3z-1)
$ of the entire expression above results to
\begin{align*}
(3z-1)(2z-1)
.\end{align*}
Substituting back $z=
x+5,$ the expression above is equivalent to
\begin{align*}
&
(3(x+5)-1)(2(x+5)-1)
\\&=
(3x+15-1)(2x+10-1)
\\&=
(3x+14)(2x+9)
.\end{align*}
