Past exam of the options II course of the University of Oxford / 2013 / P3 / b

The weight is even, so opposite-parity polynomials are automatically orthogonal. Set $p_0=1$, $p_1=x$, then subtract lower-degree components within each parity class. The needed moments areThus $p_2=x^2-{\mu }_2/{\mu }_0=x^2-2/5$ is orthogonal to $p_0$, and $p_3=x^3-({\mu }_4/{\mu }_2)x=x^3-9x/14$ is orthogonal to $p_1$. Parity supplies every other pair. One valid answer isAll have the required degree and positive norm; arbitrary nonzero rescalings are equally valid.