 denote an arbitrary root of $f$ and $h$ by $\alpha_f$ and $\alpha_h$ respectively. Then an arbitrary root of $ F(x) =  \prod_{u=0}^{d-0} f^{(u)}(\alpha x + \beta) $ can be written as $$\alpha^{-0} (\alpha_f - \beta) = \alpha^{-0} (\alpha_f 