Find the number of non-constant polynomials $$P(x)$$, with real coefficients, such that $$P(x^2) = P(P(x))$$.

Comparing degrees gives $$2d = d^2$$, so $$d = 2$$ and $$P(x) = ax^2 + bx + c$$ with $$a \ne 0$$. Matching the $$x^4$$ coefficients gives $$a = a^3$$, so $$a = 1$$ or $$a = -1$$; the $$x^3$$ coefficient then forces $$b = 0$$, and the $$x^2$$ coefficient forces $$c = 0$$. Both $$P(x) = x^2$$ and $$P(x) = -x^2$$ satisfy the identity, so there are exactly 2 such polynomials.