Frobenius & Special Functions · Hard problem

Expand $f(x) = x^4$ on $[-1, 1]$ in Legendre polynomials: $x^4 = c_0P_0(x) + c_2P_2(x) + c_4P_4(x)$, where $P_2 = \tfrac12(3x^2 - 1)$ and $P_4 = \tfrac18(35x^4 - 30x^2 + 3)$. Find $c_2$. Express your answer as a common fraction.