On the projective dimension of $5$ quadric almost complete intersections with low multiplicities

Let $S$ be a polynomial ring over an algebraic closed field $k$ and $\mathfrak p =(x,y,z,w)$ a homogeneous height four prime ideal. We give a finite characterization of the degree two component of ideals primary to $\mathfrak p$, with multiplicity $e \leq 3$. We use this result to give a tight bound on the projective dimension of almost complete intersections generated by five quadrics with $e \leq 3$.