Answer
$x^8+8x^6+24x^4y^2+32x^2y^3+16y^4$
Work Step by Step
According to the Binomial Theorem or Binomial expansion, the expansion for $(x+y)$ in the any number of power can be calculated as:
$(x+y)^n=\displaystyle \binom{n}{0}x^ny^0+\displaystyle \binom{n}{1}x^{n-1}y^1+........+\displaystyle \binom{n}{n}x^0y^n$
Let's see how this formula works:
$(x^2+2y)^4=\displaystyle \binom{4}{0}(x^2)^4(2y)^0+\displaystyle \binom{4}{1}(x^2)^3(2y)^1+\displaystyle \binom{4}{2}(x^2)^2(2y)^2+\displaystyle \binom{4}{3}(x^2)^1(2y)^3+\displaystyle \binom{4}{4}(x^2)^0(2y)^4$
or, $=x^8+4(x^6)(2y)+6(x^4)(4y^2)+16y^4$
or, $=x^8+8x^6+24x^4y^2+32x^2y^3+16y^4$
