A question that asks students to determine the quadrature nodes (and possibly weights) that maximize the order of a quadrature formula, typically involving orthogonal polynomials.
We consider the case where $I = [-1,1]$ and $w(x) = 1$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$. Deduce explicitly a quadrature formula of order 5 (you will determine the points $x_j$ and the coefficients $\lambda_j$).