Polynomials that preserve nonnegativity

Such a polynomial may be represented as $q^2 +xr^2 $ (proof: this is true for all irreducible divisors which have degree 1 or 2, and the set of polynomials of this kind is a multiplicative monoid by Euler identity.)

Strictly positive for $x>0$ polynomials may be represented as $q(x)/(x+1)^n$, where $q$ has non-negative coefficients.