Let $P(x)$ be an integer polynomial of degree $6$ that is irreducible over the integers.
$P(x) = x^6 + (A+a) x^5 + (B+ aA+ b) x^4 + (C+aB+bA +c) x^3 + (aC +bB +cA) x^2 + (bC+cB) x + cC = x^6 + (A'+a') x^5 + (B'+ a'A'+ b') x^4 + (C'+a'B'+b'A' +c') x^3 + (a'C' +b'B' +c'A') x^2 + (b'C'+c'B') x + c'C'$
Where $a,b,c,A,B,C,a',b',c',A',B',C'$ are algebraic integers of degree at most $2$.
Notice $P(x)=(x^3+ax^2+bx+c)(x^3+Ax^2+Bx+C))=(x^3+a'x^2+b'x+c')(x^3+A'x^2+B'x+C')$
I am of course intrested in two distinct factorizations of $P(x)$ so trivial solutions such as $a = a' , b = b' , c=c', ...$ are not what I seek.
Do such $P(x)$ exist ? How many exist ?