In the question Examples of nice families of irreducible polynomials over Z, user trew mentions a family of irreducible polynomials over the integers of the following form: $$ p(x) = x^4 \prod_{i=1}^{n-4} (x - b_i) - (-1)^n (2x + 4), \quad b_i \in \mathbb{Z}, \text{ pairwise distinct}, b_i \neq -2. $$ He attributes it to Philipp Furtwängler but provides no reference, and up to now I have not been able to locate it or to come up with a proof of irreducibility myself.
Here is Furtwängler's publication list on zbMathOpen: https://www.zbmath.org/?q=ai%3Afurtwangler.philipp
This family can be interpreted as a small perturbation of a reducible polynomial with real roots and experiments indicate to me that many other polynomials of such a form are irreducible (knowing though that irreducibility is generic I am not sure how much that really says...).
The polynomials appearing in the following article are of a similar form and also suspected to be irreducible:
REHMANN, U., VINBERG, E. ON A PHENOMENON DISCOVERED BY HEINZ HELLING. Transformation Groups 22, 259–265 (2017). https://doi.org/10.1007/s00031-017-9416-y.
Note: this is a repost from my MSE question https://math.stackexchange.com/questions/4099052/furtw%c3%a4nglers-family-of-irreducible-polynomials-is-there-a-perturbation-criteri
