The surface area of a right cylinder is the sum of the areas of the lateral face and the circular bases. The lateral face is really just a rectangle with the same height as the cylinder and a length equal to the circumference of the circular base.
If the radius of the base is [tex]r[/tex] and the height is [tex]h[/tex], then the surface area of the cylinder is
[tex]A=2\pi r^2+2\pi rh[/tex]
You're given that the area is [tex]96\pi[/tex] and the radius is [tex]4[/tex], so you have
[tex]96\pi=2\pi\times4^2+2\pi\times4h\implies 12=4+h\implies h=8[/tex]
