Generic factorization of a one-parameter family of degree four polynomials

Question

Let $P$ be a monic polynomial of degree four with integer coefficients. Is it true that there are only finitely many $a$ such that $P(x)-a$ factors (over $\mathbb Q$) as a product of two irreducible quadratic polynomials ? An optimistic guess would be that there is even a universal constant $C$ such that there are at most $C$ such values for $a$.

Related : Is a linear factor more likely than a quadratic factor?

Answer 1

For arbitrary natural $n$

$$x^4+4n^4=(x^2+2nx+2n^2)(x^2-2nx+2n^2)$$