I would like to know if anything can be said about the number of roots of a polynomial whose coefficients depend on the $x$, particularly, $$x^2(f(x))^2-2xf(x)+g(x)=0$$
We further know that $f(x)$ and $g(x)$ are positive for all $x$.
From Descartes' rule of signs we know that there is no negative root and that there are at most 2 positive real roots. Can we say something more? Can we say under which conditions that polynomial has exactly one (double) real root?