Answer
$(3m-5n)^2$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
9m^2-30mn+25n^2
,$ use the factoring of trinomials in the form $ax^2+bx+c.$
$\bf{\text{Solution Details:}}$
In the trinomial expression above, $a=
9
,b=
-30
,\text{ and } c=
25
.$ Using the factoring of trinomials in the form $ax^2+bx+c,$ the two numbers whose product is $ac=
9(25)=225
$ and whose sum is $b$ are $\left\{
-15,-15
\right\}.$ Using these two numbers to decompose the middle term results to
\begin{array}{l}\require{cancel}
9m^2-15mn-15mn+25n^2
.\end{array}
Using factoring by grouping, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(9m^2-15mn)-(15mn-25n^2)
\\\\=
3m(3m-5n)-5n(3m-5n)
\\\\=
(3m-5n)(3m-5n)
\\\\=
(3m-5n)^2
.\end{array}
