Let $C$ be a quartic plane curve. Suppose that for a given coordinate system
$C=(F_4(x_0,x_1,x_2)=0)$ where there the polynomial $F_4$ factorizes as

$$
F_4(x_0,x_1,x_2) = F_3(x_0,x_1,x_2)F_1(x_0,x_1,x_2)-F_2(x_0,x_1,x_2)^2
$$
If the coordinate system is fixed. How many factorization are there?

More generally, Let $Hilb_d \cong \mathbb{P}^{Nd}$ be the Hilbert scheme of curves of degree $d$ in $\mathbb{P}^2$. We can define the map $$ Hilb_3 \times Hilb_1 \times Hilb_2 \to Hilb_4 $$ as $$ \{ F_3(x_0,x_1,x_2), F_2(x_0,x_1,x_2), F_1(x_0,x_1,x_2) \} \to F_3F_1-F_2^2 $$ What can I say about the fibers of the map?

I noticed that if $F_1(x_0,x_1,x_2)$ is a bitangent of the curve $C$, then I can start playing around... However, this seems to be an elementary question, and maybe I am missing something obvious.