I looking for references to past results related to the following, probably classical, problem.
What conditions on a real polynomial of degree $n > 2$ are necessary and sufficient to conclude that it factorizes into the product of real factors of degrees $n_1,n_2,\dots,n_m$?
For example, say I have a sextic equation, $$ p(x) = Ax^6 + Bx^5 + Cx^4 + Dx^3 + Ex^2 + Fx + G. $$
I want to find conditions on $A,B,\dots,G$ such that I can be certain that $p(x)$ is the product to 2 real cubic equations (not necessarily finding the factorization of course). By the fundamental theorem of algebra, we know that there are no conditions on $A,\dots,G$ to conclude that it factorizes as the product of 3 real quadratics. But it seems nontrivial to ask when it will be the product of a 2 cubics (or maybe it is! Just looking for references or a solution if its actually an easy question). Similar would be the question of when it has 6 real roots or 6 real linear factors.