We investigate the interaction of correlated electrons with acoustical phonons using the extended Hubbard-Holstein model. The Lang-Firsov canonical transformation allows one to obtain mobile polarons for which a new diagram technique and generalized Wick's theorem is used. The physics of emission and absorption of the collective phonon-field mode by the polarons is discussed in detail. In the strong-coupling limit of the electron-phonon interaction, and in the normal as well as in the superconducting phase, chronological thermodynamic averages of products of acoustical phonon-cloud operators can be expressed by the products of one-cloud operator averages. While the normal one-cloud propagator has the form of a Lorentzian, the anomalous one is of Gaussian form and considerably smaller. We have established the Mott-Hubbard and superconducting phase transitions in this model.