In relation to my question Expression for the sum of square roots of zeros of a polynomial

How to characterize polynomials $f(x)$ with rational coefficients such that $f(x^2)=g(x)\cdot g(-x)$ where $g(x)$ is also a polynomial with rational coefficients?

Is there a computationally efficient way to identify if given polynomial $f(x)$ is such without factoring $f(x^2)$ ?