The statement is
Every degree $4$ polynomial with real coefficients is expressible as the product of two degree $2$ polynomials with real coefficients.
This and much more general versions are of course simple consequences of the Fundamental Theorem of Algebra and there are of course numerous classical ways to even calculate all the roots of the quartic, thereby establishing much stronger statements constructively.
However, suppose that we are only interested in proving the existence of such a factorisation and do not care about what the quadratics are. What's more, we want to do this for quartics only and there is no need for the argument to be applicable to polynomials of higher even degree.
Do we still need an instance of the FTA and or a classical method that does this by working out all the coefficients? Or is there now a more elementary (perhaps purely existential) argument? Perhaps one justifying the expressibility of the quartic as the difference of two polynomials squared?
This question is inspired by a similar question asked by a high school student and I am actually wondering if there is a proper proof of the above statement that is within the grasp of a high school student, i.e. no FTA, in fact no complex numbers at all, and no extensive calculations of roots/coefficients (and setting up four nonlinear equations simply by comparing coefficients falls into this class).