Method of moment generating function with Gamma Distribution

Assume that is a sample space of size n from a gamma distribution population with and unknown.

Us the method of moment generating function to show that is a pivotal quantity and has a distribution with 4n df.

I know that the moment generating function of a Gamma Distribution is .

I attempted to replicate http://www.mathhelpforum.com/math-he...-function.html, but got nowhere. I figure it has to like:

since that's the mgf of a distribution with 4n df.