In relation to my question https://mathoverflow.net/questions/123058/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 $f(x^2)$ ?