To show Gal(L/Q) is D_4, you need to verify that the resolvent cubic of f(x) factors as linear times irreducible quadratic and f(x) is irreducible over \(\displaystyle \mathbb{Q}(\sqrt{D})\), where D is the discriminant of the resolvent cubic of f(x). See here or Dummit and Foote's p613-615.