We prove that the chain polynomial of the lattice of flats of a paving matroid is real-rooted, and we define a class of matroids called generalized paving matroids. Generalized paving matroids associated to subspace lattices are shown to have real-rooted chain polynomials, by a study of a q-analog of the subdivision operator. We finish by studying single element extensions, and prove that the chain polynomials of the lattice of flats of single element extensions of (Formula presented.) and (Formula presented.) are real-rooted.