Prove that a polynomial is irreducible
Suppose that $f(x,1)=ax^3+bx^2+cx+d$ is irreducible polynomial, with integer coefficient and such that $ \gcd(a,b,c,d)=1$, with some root $ \theta$, i want to prove that $g(x,1)=x^3+bx^2+acx+a^2d$ is irriducibile and it has a root belonging to $ \mathbb{Q}(\theta) $. If i found a matrix in $\mathbf{GL}_2(\mathbb{Z}) $ that transform $f(x,y)$ into $g(x,y)$ i'm done, but i can't find any...
HINT:
We have $$a^2 f(x) = a^2 ( a x^3 + b x^2 + c x + d ) = (a x)^3 + b (a x)^2 + a c (a x) + a^2 d = g(a x)$$