Answer
$(4r+7q)^2$
Work Step by Step
Using the factoring of trinomials in the form $ax^2+bx+c,$ the expression
\begin{align*}
16r^2+56rq+49q^2
\end{align*} has $ac=
16(49)=784
$ and $b=
56
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
28,28
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{align*}
16r^2+28rq+28rq+49q^2
.\end{align*}
Grouping the first and second terms and the third and fourth terms, the expression above is equivalent to
\begin{align*}
(16r^2+28rq)+(28rq+49q^2)
.\end{align*}
Factoring the $GCF$ in each group results to
\begin{align*}
4r(4r+7q)+7q(4r+7q)
.\end{align*}
Factoring the $GCF=
(4r+7q)
$ of the entire expression above results to
\begin{align*}
&
(4r+7q)(4r+7q)
\\&=
(4r+7q)^2
.\end{align*}
Hence, the factored form of $
16r^2+56rq+49q^2
$ is $
(4r+7q)^2
$.
