#### Answer

$(5r-9)^2$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To factor the quadratic expression $ax^2+bx+c,$ find two numbers whose product is $ac$ and whose sum is $b$. Use these $2$ numbers to decompose the middle term of the quadratic expression and then use factoring by grouping.
$\bf{\text{Solution Details:}}$
In the given expression, $
25r^2-90r+81
,$ the value of $ac$ is $
25(81)=2025
$ and the value of $b$ is $
-90
.$ The $2$ numbers that have a product $ac$ and a sum of $b$ are $\{
-45,-45
\}.$ Using these $2$ numbers to decompose the middle term of the given expression results to
\begin{array}{l}\require{cancel}
25r^2-45r-45r+81
.\end{array}
Grouping the first and third terms and the second and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
(25r^2-45r)-(45r-81)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
5r(5r-9)-9(5r-9)
.\end{array}
Factoring the $GCF=
(5r-9)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(5r-9)(5r-9)
\\\\=
(5r-9)^2
.\end{array}