Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial. Consider $\bar{f} \in \mathbb{F}_p[x].$ Let $\rho_p$ be the number of distinct roots of $\bar{f}$ in $\mathbb{F}_p$, and let $\rho$ be the number of distinct roots of $f$ in $\mathbb{C}$. Do you have a reference for showing that $\rho_p = \rho$ for infinitely many primes $p$?

My background to this question: I actually use this result to calculate the precise maximal subgroup growth of groups of the type $\mathbb{Z}^k \rtimes \mathbb{Z}$ (and of similar metabelian groups).