Let F(x) be a polynomial with real coefficients. Find all functions F(x) satisfying:

F(x)*F(2x^2-1) = F(x^2)*F(2x-1)

If the highest degree term is a.z^n, then

F(x) has a.x^n

F(2x^2-1) has a.2^n.x^2n

So the left hand side has a^2.2^n.x^3n

F(x^2) has a.x^2n

F(2x-1) has a.2^n.x^n

So the right hand side has a^2.2^n.x^3n

A priori, the problem seems possible. At least, f(x)=C for constant C, does the job.