A recent question, Is irreducibility of polynomials $\in\mathbb{Z}[X]$ over $\mathbb{Q}$ an undecidable problem? was quickly answered in the negative. I am wondering if there is a simple example of a family of families of integer polynomials whose irreducibility is undecidable. For example, consider the following computational problem:

Instance: A positive integer $n$.
Question: Does the family of polynomials $\{x^d + x + n : d \in \mathbb{N}\}$ contain infinitely many members that are irreducible over $\mathbb{Q}$?

I don't know off the top of my head whether the above computational problem is undecidable. If it is, then that would answer my question affirmatively. If not, or if its undecidability is unknown, then is there some other problem of comparable simplicity that we can prove is undecidable?