Given a degree $d$, it is possible to construct a pair $(F,\delta),$ where $F$ is a polynomial in $\mathbb{Z}[X]$ and $\delta$ a non-zero integer, such that $F(X)$ and $F(X)+\delta$ both split into linear factors over $\mathbb{Z}[X]$ ?
This is easy for $d=2$, as shown by the pair $F(X)=X^2$ and $\delta=-a^2$ (for an arbitrary integer $a$). Indeed, $F(X)=X \cdot X$ and $F(X)+\delta=(X-a)\cdot (X+a).$ What can be said for $d>2$ ?