Let's define polynomial $P(x)$ as $ P(x)=x^3+2x^2-4x+2 $. My main question is to prove that it is irreducible and that its discriminant is $-140$, from the definition of polynomial discriminant and using cubic method of discriminant. Here is a picture:
$D_3=a_1^2a_2^2-4a_0a_2^3-4a_1^3a_3+18a_0a_1a_2a_3-27a_0^2a_3^2$
How can we show that it is irreducible? Thanks a lot.