Show polynomial is irreducible Show that $x^4 + 4x^3 + 6x^2 + 2x + 1$ is irreducible over the field of rational numbers Q. 
Now I've tried substituting x-1 for x to show that it is irreducible, but I'm having a hard time. This is the correct way to go about it, I know, but I don't know how to execute the problem.
Let $P(x)=x^4+4x+6x^2+2x+1$ then $P(x)=(x+1)^4-2x=(x+1)^2-2(x+1)+2=Q(x+1)$, with $Q(y)=y^4-2y+2$. Now Eisenstein criterion applies with $p=2$.
Just take the polynomial modulo $2$. By the RRT, $x^4+1$ is irreducible so we are done.