Degree of Precision and Error Term of a Quadrature Rule via Taylor Expansion

A quadrature rule $\int_a^b f \approx Q$ has degree of precision $d$ if it is exact for every polynomial of degree $\le d$ but not for degree $d+1$. Taylor-expanding both sides about a point and matching terms reveals $d$ and the leading error: the first mismatched power gives an error $\propto h^{d+2} f^{(d+1)}(\xi)$.