This extended abstract is a summary of a recent paper which studies the enumeration of faces of subdivisions of cell complexes. Motivated by a conjecture of Brenti and Welker on the real-rootedness of the h-polynomial of the barycentric subdivision of the boundary complex of a convex polytope, we introduce a framework for proving real-rootedness of h-polynomials for subdivisions of polytopal complexes by relating interlacing polynomials to shellability via the existence of so-called stable shellings. We show that any shellable cubical, or simplicial, complex admitting a stable shelling has barycentric and edgewise subdivisions with real-rooted h-polynomials. Such shellings are shown to exist for well-studied families of cubical polytopes, giving a positive answer to the conjecture of Brenti and Welker in these cases. The framework of stable shellings is also applied to answer to a conjecture of Mohammadi and Welker on edgewise subdivisions in the case of shellable simplicial complexes.