How to show a polynomial is reducible/irreducible in a ring

Question

I have to show that these polynomials is reducible or irreducible in the given ring.

$a)$ $2x^3 − 5x^2 + 6x − 2$ in $\mathbb{Z}[x]$

$b)$ $x^4 + 4x^3 + 6x^2 + 2x + 1$ in $\mathbb{Z}[x]$

I think I have to use Eisenstein's Theorem for these choosing a prime number $p$ but I'm not very good at choosing the right $p$ and where to go from there. Also for $b)$ I think substituting $x$ for $x-1$ would be easier?

Answer 1

We have $$ 2x^3-5x^2+6x-2=(x^2-2x+2)(2x-1), $$ so that this polynomial is reducible. The second one is irreducible with $p=2$ by Eisenstein, for $f(x-1)$. The transformed polynomial is $x^4-2x+2$.