The problem is stated as follows:
Let $F$ be an affine plane curve. Let $L$ be a line that is not a component of $F$. Suppose $L = \{(a+tb, c +td)\, | \, t \in k \}$. Define $G(T) = F(a+Tb, c+Td)$. Factor $G(T) = \epsilon \prod (T-\lambda_i)^{e_i}$, $\lambda_i$ distinct. Show that there is a natural one-to-one correspondance between the $\lambda_i$ and the points $P_i \in L \cap F$. Show that under this correspondance, $I(P_i, L \cap F) = e_i$. In particular, $\sum I(P_i, L \cap F) \leq \deg (F)$.
I was able to solve the first part, the correspondece between roots of $G$ and the intersection points of $L\cap F$ is given by $P_i = (a+\lambda_i b, c+\lambda_i) d$. But I can't finish the question. I don't know how to find a useful relation between $L$, $F$ and $G$.
An idea could be to show that $ord_{P_i}^L(F) = e_i$, because it coincides with the intersection number by a previous exercise.
Thanks in advance.
For each $P_i$, you can construct the affine change of coordinates which maps $(0,0)$ to $P_i$ and X to L. So, you may assume that $L=\{(0,t): t\in k\}$ and $G(T)=F(0,T)$.
Thus $F(0,T)=T^{e_i}(1+H(T))$ and $F(X,Y)=Y^{e_i}(1+H(Y))+X(B(X,Y))$.
From the properties of intersection number, $I(P,X\cap F)=I(P,X\cap Y^{e_i})=e_i$.