I have the specific function $G$ of $x$ satisfying the functional equation $G^3x=-1+G(1-x)+G^2x$. By using Mathematica, I would like to simplify the following polynomial of $G$ and $x$.
$2 G^2 + 6 G^2 x + 4 G^3 x + 3 G^2 x^2 - 8 G^3 x^2 - 2 G^4 x^2$
Because of the given relation, we should be able to substitute $G^3x$, and express the above as the polynomial with $G^n$, $x^i$, $Gx^j$, and $G^2x^k$ terms. Mathematica does not allow the assignment of a power, so I cannot figure out the reasonable way of doing this.
Is there any efficient method to solve this by using Mathematica?