Determining structure factors for crystals from lattice and motif

I understand that the structure factor for a crystal is given by:

$$F_{hkl} = \sum_r f_{r} e^{2{\pi}i(hx_{r}+ky_{r}+lz_{r})} $$

My textbook (''Structures of crystals'', Glazier) suggests that you can calculate structure factors if you know the lattice type and motif of the crystal, simply by finding the product of the lattice and the motif structure factors.

Can this method be used generally, and if so, do you need to define the motif relative to a centre of symmetry in order to get real structure factors for centrosymmetric crystals ?