 Solution

Use the intended degree ordering from part (b). Let ${\ell }_j$ be the degree-$n$ Lagrange cardinal polynomial at the distinct roots $x_0,\dots ,x_n$ of $p_{n+1}$:For any $f\in {\Pi }_{2n+1}$, polynomial division gives $f=s{p}_{n+1}+r$ with $\mathrm{deg}s\le n$ and $\mathrm{deg}r\le n$. Orthogonality of $p_{n+1}$ to ${\Pi }_n$ makes the integral of $s{p}_{n+1}$ zero, and that product also vanishes at every quadrature node. Meanwhile $r={\sum }_{j}r(x_j){\ell }_j$ by interpolation. ConsequentlyThus the rule is exact up to degree $2n+1$. To determine the sign of a weight, apply exactness to ${\ell }_j^2$, of degree $2n$:The inequality is strict because ${\ell }_j$ is a nonzero polynomial and the weight is positive. This is the Gaussian quadrature construction; the source asks only for nonnegative weights, and the proof gives positivity.