 Two-node Gaussian quadrature with quadratic weight

The even weight $1-x^2$ has moments $m_0=4/3$ and $m_2=4/15$. Its monic quadratic orthogonal polynomial is $x^2-1/5$, so the two Gaussian quadrature nodes are $\pm 1/\sqrt{5}$. Symmetry and constant exactness give weights $2/3$ each. These nodes and weights integrate every polynomial of degree at most three exactly. Degree four fails: the true integral is $4/35$, whereas the rule gives $4/75$.