I need to show that $x^4+4x^3+6x^2+9x+11$ is irreducible in the integers.
First, I tried to apply Eisenstein's irreducibility criterion by shifting $x$ to $x+\alpha$. However, I can't think of any shift to apply that would fit the criterion.
Next, I tried using polynomial division. If it were reducible, this polynomial would have either a linear or a quadratic factor. It has no integer roots, so I tried to divide by a factor $x^2+ax+b$ and require that the remainder be zero; however, this yields two very difficult equations, I'm not sure how to prove that they have no integer solution.
Any hints?