#### Answer

$(5z-3m)^2$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To factor the given expression, $
25z^2-30zm+9m^2
,$ find two numbers whose product is $ac$ and whose sum is $b$ in the quadratic expression $ax^2+bx+c.$ Use these $2$ numbers to decompose the middle term of the given quadratic expression and then use factoring by grouping.
$\bf{\text{Solution Details:}}$
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
25(9)=225
$ and the value of $b$ is $
-30
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
-15,-15
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
25z^2-15zm-15zm+9m^2
.\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}
(25z^2-15zm)-(15zm-9m^2)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
5z(5z-3m)-3m(5z-3m)
.\end{array}
Factoring the $GCF=
(5z-3m)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(5z-3m)(5z-3m)
\\\\=
(5z-3m)^2
.\end{array}