Prove that $x^3+12x^2+18x+6$ is irreducible over $\mathbb{Z}[i]$.

Question

Prove that $x^3+12x^2+18x+6$ is irreducible over $\mathbb{Z}[i]$.

Note: It has been a minute since I have done a problem like this.

I am told to use Eisenstein's Criterion. So I need to find a prime, p, that divides 12, 18, and 6 but $p^2$ doesn't divide 6. Well I can see that p can be either 2 or 3. Does it matter whether I use 2 or 3?

Also is this enough for the proof?