Fulton's Algebraic Curves makes us deal with the following exercise (ex. 3.21):
3.21. 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 $G(T)=\epsilon\Pi(T-\lambda_i)^{e_i}, \lambda_i$ distinct. Show that there is a natural one-to-one correspondence between the $\lambda_i$ and the points $P_i\in L\cap F$. Show that under this correspondence, $I(P_i,L\cap F)=e_i$. In particular, $\Sigma I(P,L\cap F)\leq \operatorname{deg}(F)$.
Algebraic Curves, Fulton (2008).
I think I have come with a possible solution; I just need to handle a simple step for which I need some help.
Of course, the correspondence is trivially $\lambda_i \leftrightarrow \left (a+\lambda_ib,c+\lambda_id\right )=:P_i$.
Fix $i$. If we make an appropriate linear change of coordinates, we can make the assumption that $L=y$ and $P_i=\left (0,0\right )$. And now it is very easy. Just write $F=F(x,0)+yG$, and we know that $F(x,0)\neq 0$ since $y\nmid F$. Now $I(P_i,F\cap L)=I((0,0),F(x,0)\cap y)$ and this is clearly equal to the multiplicity of $0$ as a root of $F(x,0)$.
Since $L=y$, then $(a+tb,c+td)=(t,0)$, therefore $\lambda_i=0$ and $G(t)=F(t,0)$, so we have our desired equality.
And we are done... but I have a little doubt. How do we know that linear changes of coordinates preserve the multiplicities? Moreover: what is the relationship between the original $\lambda_i$ and the new one under an arbitrary linear change of coordinates? I am not succeeding in understanding completely this. Any help?
If you take the definition of intersection multiplicity of two affine curves given by $V(f),V(g)$ at a point $p$ as the length of the module $\mathcal{O}_{\Bbb A^2,p}/(f,g)$, then this is an isomorphism invariant by the Jordan-Holder theorem. As for the how the $\lambda$ change under the coordinate change: they don't. The vectors which determine the line are transformed by the linear change of coordinates, but the $\lambda$ don't change.