Fix a nonzero integer $a$ and a positive integer $k$. I'm looking for some criterion to establish for which nonzero integers $b$ and $n \geq 2$ the polynomial $$f(x) := x^n - (x+a)^{n-k} (x+b)^k$$ is irreducible over $\mathbb{Z}[x]$. I think it is irreducible at least for all sufficiently large $n$, but I failed to prove it.
EDIT: The question is more complicated than I expected. So I specify that I'd be happy just to prove that there are infinitely many $n$ such that $f(x)$ is irreducible in $\mathbb{Z}[x]$ for any nonzero integer $b \neq -a$. Peter Mueller's comments below show that these $n$ must be relatively prime with $k$, so maybe a good guess is to consider $n$ prime.
Thank you in advance for any idea.