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A mixed spin-1 and spin-2 Ising ferrimagnetic system is investigated on a square lattice of spins S = ± 2 , ± 1 , 0 and Q = ± 1 , 0 in the presence of an h ′ external magnetic field, such that S and Q are nearest-neighbors antiferromagnetically coupled. Using Monte Carlo simulations we calculate the finite temperature phase diagrams of the magnetization and the specific heat. We consider ferromagnetic interactions between next-nearest neighbors, S spins ( J 2 ′ ), as well as between the Q spins ( J 3 ′ ), and crystal field and external magnetic field interactions at each point of the lattice. Under these conditions the system exhibits spin compensation temperatures and a first-order reentrant behavior in a certain range of the parameters of the Hamiltonian. We analyze the behavior of the critical and compensation temperatures, as well as the first order phase transition temperatures, with the external magnetic and crystal fields, and the next-nearest neighbors interaction between S spins. We found that the existence of compensation temperatures and a first order phase transition depends on the strength of h ′ , J 2 ′ and D ′ . [ABSTRACT FROM AUTHOR]