Is the cubic polynomial $x^3-ax-(2a+1)$ irreducible over $\mathbb Z$ for all positive integers $a$?
one way to prove irreducibility is to use eisenstein's criterion. we want to find prime $p$ st:
However, the first 2 conditions imply $p\mid 1$, a contradiction. so this approach fails.
another approach is to use the rational root thm.
where b,c,d integers
there was a link to a similar problem, now deleted.