 Solution

Factor $f_1=(X^2-3)(X^2+2)$. Its splitting field is $L=\mathbb{Q}(\sqrt{3},\sqrt{-2})$. The first quadratic field is real, so it cannot contain the nonreal $\sqrt{-2}$. Thus adjoining the second square root doubles the degree again. Independent sign changes of the two square roots give all four automorphisms, so