Answer
$(4r+5)^2$
Work Step by Step
Let $z=(4r+1)$. Then the given expression, $
(4r+1)^2+8(4r+1)+16
$, is equivalent to
\begin{array}{l}
z^2+8z+16
.\end{array}
The two numbers whose product is $ac=
1(16)=16
$ and whose sum is $b=
8
$ are $\{
4,4
\}$. Using these two numbers to decompose the middle term of the expression, $
z^2+8z+16
$, results to
\begin{array}{l}
z^2+4z+4z+16
\\\\=
(z^2+4z)+(4z+16)
\\\\=
z(z+4)+4(z+4)
\\\\=
(z+4)(z+4)
\\\\=
(z+4)^2
.\end{array}
Since $z=(4r+1)$, then,
\begin{array}{l}
(z+4)^2
\\\\=
(4r+1+4)^2
\\\\=
(4r+5)^2
.\end{array}
