Answer
$(9z+2)^2$
Work Step by Step
Using the factoring of trinomials in the form $ax^2+bx+c,$ the given expression
\begin{align*}
81z^2+36z+4
\end{align*} has $ac=
81(4)=324
$ and $b=
36
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
18,18
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{align*}
81z^2+18z+18z+4
.\end{align*}
Grouping the first and second terms and the third and fourth terms, the expression above is equivalent to
\begin{align*}
(81z^2+18z)+(18z+4)
.\end{align*}
Factoring the $GCF$ in each group results to
\begin{align*}
9z(9z+2)+2(9z+2)
.\end{align*}
Factoring the $GCF=
(9z+2)
$ of the entire expression above results to
\begin{align*}
&
(9z+2)(9z+2)
\\&=
(9z+2)^2
.\end{align*}
