 Solution

Odd terms vanish and direct evaluation gives $L(p)=(8/45)c_4$, so the rule is exact through degree $3$. With $n=3$, the Peano kernel isIts nonnegativity and $\int K=L(x^4)/4!=1/135$ give $|L(f)|\le \Vert f^{(4)}{\Vert }_{\infty }/135$, so $N=135$. Equality holds for a quartic with fourth derivative of constant sign.