In 5.23., Fulton proves the theorem that if $P$ is an ordinary flex of a plane curve $C=V(F)$, then $C$ and it's Hessian $H$ intersect with multiplicity one, that is $I(P, C \cap H) = 1$.
After a bit of work, we have that it suffices to check the intersection multiplicity of the curves $f = F(X,Y,1) = y + ax^2 + bxy + cy^2+...$ and $g = 2a + 6dx + ...$ in the point $P = [0:0:1]$. Here, $a = 0$ and $d \neq 0$. What I don't see is how this implies the statement. Why do we get $\dim_k \mathcal{O}_0(\mathbb{A}^2) /(f, g) = 1$ from this? I have so far only computed multiplicities when the ideals were just monomials, so I'd appreciate any help.