Let $0<m<n$ in $\mathbb{N}$ and $a_i,b_j\in\mathbb{C}$ for $0\leq i\leq m$ and $0\leq j\leq n$ and $a_m,b_n\neq 0$.
Consider
$(\sum_{i=0}^{m}a_i (X+Y)^i)(\sum_{j=0}^{n}b_i (1+Y)^i)(\sum_{j=0}^{n}a_i Y^i)+(\sum_{i=0}^{m}a_i (1+Y)^i)(\sum_{j=0}^{n}b_i (X+Y)^i)(\sum_{j=0}^{n}a_i Y^i)-2(\sum_{i=0}^{m}a_i Y^i)(\sum_{j=0}^{n}b_i (X+Y)^i)(\sum_{j=0}^{n}a_i (1+Y)^i)$
as a polynomial $P_{X}(Y)$ in the variable $Y$.
Let $A$ be the set of all $a\in\mathbb{C}$ such that $P_{a}(Y)$ is separable. It is clear that either $A$ is cofinite or empty.
I want to prove that $A$ is cofinite. I would be very thankful for any idea or counterexample.