#### Answer

$(4a+5b)^2$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To factor the given expression, $
16a^2+40ab+25b^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 $
16(25)=400
$ and the value of $b$ is $
40
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
20,20
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
16a^2+20ab+20ab+25b^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}
(16a^2+20ab)+(20ab+25b^2)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
4a(4a+5b)+5b(4a+5b)
.\end{array}
Factoring the $GCF=
(4a+5b)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(4a+5b)(4a+5b)
\\\\=
(4a+5b)^2
.\end{array}