Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / P / P3 / d

By part (c), the best degree-at-most-two polynomial is odd, so it must be $p_2(x)=bx$. Orthogonality to $x$ givesThereforeIts error is automatically orthogonal to the even basis functions $1,x^2$ by parity, and the chosen coefficient makes it orthogonal to $x$, so part (b) verifies optimality.