Answer
$(5r+2s-3)^2$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
(5r+2s)^2-6(5r+2s)+9
,$ use substitution. Then use the factoring of trinomials in the form $x^2+bx+c.$
$\bf{\text{Solution Details:}}$
Let $z=(5r+2s).$ Then the given expression is equivalent to
\begin{array}{l}\require{cancel}
z^2-6z+9
.\end{array}
In the trinomial expression above, $b=
-6
,\text{ and } c=
9
.$ Using the factoring of trinomials in the form $x^2+bx+c,$ the two numbers whose product is $c$ and whose sum is $b$ are $\left\{
-3,-3
\right\}.$ Using these two numbers, the factored form of the expression above is
\begin{array}{l}\require{cancel}
(z-3)(z-3)
\\\\=
(z-3)^2
.\end{array}
Since $z=(5r+2s),$ by back substitution, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(5r+2s-3)^2
.\end{array}