Polynomials: gaining irreducibility by adding a constant EDITED:
Let $f\in\mathbb{Q}[X]$. I'm interested in the factoring properties of the family $F:=\{f+c\}_{c\in\mathbb{Q}}$ of rational polynomials that differ from $f$ only by a constant. Specifically:
1) I think there must be at least one irreducible polynomial in $F$. How can we prove it?
2) If $f$ is reducible, can we effectively determine an specific $c$ such that $f+c$ is irreducible?
3) How are the asymptotic growth and the density of those $c$ such that $f+c$ is irreducible? Are there at least an infinite number of them? How does this depend on the coefficients of the polynomial?
E.g, if $f(X):=X^2$ then any $c>0$ makes $f(X)+c$ irreducible, but it looks like $X^3+3X$ can be made reducible by infinite (ever sparser) values of $c$.
4) If $f\in\mathbb{Z}[X]$, can something be said about the residue distribution of the $c\in\mathbb{Z}$ modulo any prime?
5) How do the results generalize to other fields? What happens with degree 2 polynomials in real closed fields? What happens exactly in fields of characteristic $p$?

The case of degree $2$ polynomials in $\mathbb{R}$ or $\mathbb{Q}$ can be handled. Let $f(x) = ax^2 + bx$. By the quadratic formula, if $b^2 - 4ac < 0$, then we have $f(x) + c$ irreducible.
