Answer
$(3a-7)^2$
Work Step by Step
Let $z=a-4.$ Then the given expression, $
9(a-4)^2+30(a-4)+25
,$ is equivalent to
\begin{array}{l}\require{cancel}
9z^2+30z+25
.\end{array}
The two numbers whose product is $ac=
9(25)=225
$ and whose sum is $b=
30
$ are $\{
15,15
\}.$ Using these two numbers to decompose the middle term, then the factored form of $
9z^2+30z+25
$ is
\begin{array}{l}\require{cancel}
9z^2+15z+15z+25
\\\\=
(9z^2+15z)+(15z+25)
\\\\=
3z(3z+5)+5(3z+5)
\\\\=
(3z+5)(3z+5)
\\\\=
(3z+5)^2
.\end{array}
Since $z=a-4$, then the expression $
(3z+5)^2
$ is equivalent to
\begin{array}{l}\require{cancel}
(3(a-4)+5)^2
\\\\=
(3a-12+5)^2
\\\\=
(3a-7)^2
.\end{array}
