Math question: closed form solution

Question for the math wizzes:

Assume
0 < e < 1
0 < p < 1
g > 0

Is there a closed-form solution to the following equation?

px^(2+2e) + g*(1-p)x^2 - gpx^(2e) - (1-p) = 0

I can find a numerical solution using Matlab, and it works just fine for my purposes, but I’m curious whether it has a closed-form solution.

It has such a “nice” look to it that I figure that there must be a way to re-arrange it and find a closed form solution to it.

Maple doesn’t find anything.

I just tried Mathematica, and it says


Solve::tdep: The equations appear to involve the variables to be solved for in
    an essentially non-algebraic way.


So, maybe it is hopeless.