The first term, $16q^2\text{,}$ is a perfect square because it equals $(4q)^2\text{.}$ So we could take $A=4q\text{.}$ The last term, $49\text{,}$ is also a perfect square since it is equal to $7^2\text{.}$ So we could take $B=7\text{.}$ Now we multiply $2AB=2\cdot(4q)\cdot7\text{,}$ and the result is $56q\text{.}$ This is the middle term, which is what we hope to see. So we can use Fact 7.5.9: $$\begin{aligned} 16q^2+56q+49\amp=(A+B)^2\\ \amp=(4q+7)^2 \end{aligned}$$