Answer
$2(x^{a}+2y^{b})(x^{2a}-2x^ay^{b}+4y^{2b})$
Work Step by Step
Factoring the $GCF=
2,
$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
2x^{3a}+16y^{3b}
\\\\=
2(x^{3a}+8y^{3b})
.\end{array}
The expressions $
x^{3a}
$ and $
8y^{3b}
$ are both perfect cubes (the cube root is exact). Hence, $
x^{3a}+8y^{3b}
$ is a $\text{
sum
}$ of $2$ cubes. Using the factoring of the sum or difference of $2$ cubes which is given by $a^3+b^3=(a+b)(a^2-ab+b^2)$ or by $a^3-b^3=(a-b)(a^2+ab+b^2)$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
2[(x^{a})^3+(2y^{b})^3]
\\\\=
2(x^{a}+2y^{b})[(x^{a})^2-x^a\cdot (2y^{b})+(2y^{b})^2]
\\\\=
2(x^{a}+2y^{b})(x^{2a}-2x^ay^{b}+4y^{2b})
.\end{array}
