Answer
$a^{2n}+8a^nb^n+16b^{2n}$
Work Step by Step
The square of a binomial sum is: $(A+B)^2=A^2+2AB+B^2$.
The square of a binomial difference is: $(A-B)^2=A^2-2AB+B^2$.
Hence here: $(a^n+4b^n)^2=(a^n)^2+2\cdot a^n\cdot 4b^n+(4b^n)^2=a^{2n}+8a^nb^n+16b^{2n}$