Answer
$x^{16}+6x^8+9$
Work Step by Step
Using $(a\pm b)^2=a^2\pm2ab+b^2$ or the square of a binomial, the square of the given expression, $\left(
x^8+3
\right)^2,$ is
\begin{array}{l}\require{cancel}
(x^8)^2+2(x^8)(3)+(3)^2
\\\\=
x^{16}+6x^8+9
.\end{array}