A general analytical expression for the electron energy spectrum and the polaron binding energy for different electronic subband bound states in a quantum well (QW) is presented. The effects of the electron-optical-phonon interaction, the finite confinement potential, the difference in the electronic effective masses across the interface, and the electron subband state are considered in this paper. The correct three-dimensional and two-dimensional results are given as the limit cases when the well width varies from infinity to zero. The expression is numerically applied to the GaAs/Al_xGa_1-xAs QWS with several different aluminium concentrations x. Some properties of the polaron are discussed.