I am reading William Fulton's Algebraic Curves. There are few properties listed which can be used to compute intersection number of two plane curves. For example, If we have plane curves $F$ and $G$ passing through $P$ that do not have common tangent at $P$, then, $I(P,F\cap G) = m_P(F)m_P(G)$.
In my mind, I have this intution , that for each pair of tangent lines of $F$ and $G$ , they intersect transversally, contributing $1$ to $I(P,F \cap G)$
That is, suppose , we have a curve $F$ with tangent line $L_1$ with multiplicity $2$ and tangent line $L_2$ with multiplicity $5$ , and $G$ has tangent line $L_3$, $L_4$ with multiplicity $1,4$ respectively at $P$, then, assuming all $L_i$ are distinct , $I(P,F\cap G)=7\cdot 5=35$.
Now, suppose $L_2 = L_4$ in the previous setting. Just with this information can we say anything about $I(P,F\cap G)?$ If not, can we say something with little additional information?