Actually I tried solving it.
A) By Eisenstein's Criteria its is irreducible.
B) and C) These polynomials don't have any zero in $\mathbb Z$ hence are also irreducible.
D) I have doubt in this tried it like ... Let $$x=y+1$$ then equation becomes $$y^4 + 5y^3+10y^2+10y+5.$$
By Eisenstein's Criterion this is also irreducible.
But its not possible, one of them is not irreducible!! Can someone help me?
$2x^3+6x+12 =2(x^3+3x+6)$
Note that $2$ is not a unit in $\mathbb Z [x]$.
The criterion you cited (that having no roots implies irreducibility for polynomials of degree $\leq 3$) works only over fields.