Roots of polynomial equations

Question

Is there anything that can be said about the roots of the polynomial $f_n(x)$ if $f_n(x) = xf_{n-1}(x) + f_{n-2}(x)$ where these are polynomials of degree $n, n-1,$ and $n-2$, respectively. In addition, the roots of both $f_{n-1}(x)$ and $f_{n-2}(x)$ are known to be the maximum number of complex conjugate pairs (at most one purely real root), and all roots have positive real part.