Answer
$4x^2+y^2+z^2=16$
Work Step by Step
$4x^2+y^2=16$
$\frac{x^2}{4}+\frac{y^2}{16}=1$
The ellipse rotated around the x-axis will have circles as parallel to the yz-plane. Thus, we have to add a $z^2$ term that has the same coefficient as the $y^2$ term.
$\frac{x^2}{4}+\frac{y^2}{16}+\frac{z^2}{16}=1$
or, $4x^2+y^2+z^2=16$
