Determine the Galois group of $f(x)$ over $\mathbb{Q}$
$f(x)=x^6+22 x^5-9 x^4+12 x^3-37 x^2-29 x-15$
This question comes from Johns Hopkins University Fall 2018 algebra qualifying.
I have found it is irreducible over rational numbers by wolfram, but have no idea how to show its irreducibility on examination(Maybe Eisentein's criterion,but there are so many terms to compute, and I think I do not have so many time to calculate on exam =_=).
The Galois group $G$ is a subgroup of $S_6$. And $f$ has two real root and hence two pairs of conjugate complex roots. Module a pirime p may work, but there are still so many terms and irreducibility to verify.
Any quick method will be appreciated.