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We investigate the effects of a two-band model in the description of the negative electronic compressibility (NEC) for the two-dimensional electron gas (2DEG). Using a homogeneous 2D interacting electron model, we examine the crossover point of the electronic compressibility of the 2DEG with one and two electron bands and find that the presence of inter-band coupling produces a dramatic decrease in the effective dielectric constant and the critical carrier density of the 2DEG. Furthermore, we investigate how these parameters change with variable effective mass. Additionally, we apply our results to the NEC observed in the 2D electron gas at the interface of LaAlO₃/SrTiO₃ (LAO/STO) and demonstrate how one can observe effects of inter-band interactions in electronic compressibility. We determine that for the known parameters of LAO/STO, the system may be a realization of two-band 2D electron gas.