Answer
$2(4a+3b)^{2}$
Work Step by Step
Factor out the greatest common factor, $2$
$32a^{2}+48ab+18b^{2}=2(16a^{2}+24ab+9b^{2})=*$
Test whether the parentheses hold a perfect square,
$(A+B)^{2}=A^{2}+2AB+B^{2}$
First term: $A^{2}=(4a)^{2}\Rightarrow A=4a$
Third term: $B^{2}=(3b)^{2}\Rightarrow B=3b$
Test:$\qquad$ does $2AB$ equal the middle term?
$ 2AB=2(4a)(3b)=24ab\qquad$ ... yes, it does.
The parentheses hold a perfect square, $(4a+3b)^{2}$
$*...= 2(4a+3b)^{2}$
