Answer
$(10x+17)^2$
Work Step by Step
Let $z=5x+7.$ Then the given expression, $
4(5x+7)^2+12(5x+7)+9
,$ is equivalent to
\begin{array}{l}\require{cancel}
4z^2+12z+9
.\end{array}
The two numbers whose product is $ac=
4(9)=36
$ and whose sum is $b=
12
$ are $\{
6,6
\}.$ Using these two numbers to decompose the middle term, then the factored form of $
4z^2+12z+9
$ is
\begin{array}{l}\require{cancel}
4z^2+6z+6z+9
\\\\=
(4z^2+6z)+(6z+9)
\\\\=
2z(2z+3)+3(2z+3)
\\\\=
(2z+3)(2z+3)
\\\\=
(2z+3)^2
.\end{array}
Since $z=5x+7$, then the expression $
(2z+3)^2
$ is equivalent to
\begin{array}{l}\require{cancel}
(2(5x+7)+3)^2
\\\\=
(10x+14+3)^2
\\\\=
(10x+17)^2
.\end{array}
