In　this　paper,　we　consider　the　compressible　Navier-Stokes-Maxwell　equations　with　a　non-constant　background　density　in　R-3.　We　first　show　the　existence　and　uniqueness　of　the　non-trivial　equilibrium　(steady-state)　of　the　system　when　the　background　density　is　a　small　variation　of　certain　constant　state,　then　we　prove　the　asymptotic　stability　of　the　steady-state　once　the　initial　perturbation　around　the　steady-state　is　small.　Furthermore,　by　establishing　the　time-decay　estimates　for　the　corresponding　linearized　homogeneous　equations,　we　artfully　derive　the　time-algebraic　convergence　rates.　The　proof　is　based　on　the　time-weighted　energy　method　but　with　some　new　developments　on　the　weight　settings.