We apply Gauss's Law. The charge contained within the sphere is
$\begin{align*}\sigma \pi \left(R^2 + x^2\right)\end{align*}$
Therefore, by Gauss's Law, the electric flux $\phi$ through the Gaussian surface is

$\begin{align*}\boxed{\frac{\sigma \pi \left(R^2 + x^2\right)}{\epsilon_0}}\end{align*}$
Therefore, answer (D) is correct.

Solution to 2001 Problem 60