Why if I show that that $f(x-1)$ is irreduction, so $f(x)$ is also irreductible?

Question

I have to show that $f(x)=x^4+4x^3+6x^2+2x+1$ is irreductible in $\mathbb{Z}[x]$. With Eisenstein's criterion, why if I show that that $f(x-1)$ is irreducible, so $f(x)$ is also irreducible? It is important to know that $f(x-1)=x^4-2x+2$ and works with Eisenstein's criterion.

Answer 1

$f(x) =g(x)h(x)$ where $g,h$ are polynomials of degree less than $\deg f$ if and only if $$f(x-1)=g(x-1)h(x-1).$$

Hence $f(x)$ is reducible if and only if $f(x-1)$ is.

Therefore, since Eisenstein's criterion tells us that $f(x-1)$ is irreducible, we deduce that $f(x)$ is irreducible.