The form of the field distribution in a random-field Ising model is known to play an important role in determining the order of the phase transition and the existence of the tricritical points. Therefore, we have investigated the trimodal random-field spin-2 Ising system in a transverse field by combining the pair approximation with the discretized path-integral representation. This is done by introducing a parameter p which simulates the fractions of the spins exposed to h(0), -h(0) and 0 external longitudinal magnetic fields. The variation of the critical reduced transverse and longitudinal magnetic fields with the parameter p is studied for different coordination numbers, i.e., z = 4, 6, 8, 12 and infinity (leads to the mean-field results), and it is found that the system exhibits tricritical points for only p <= 0.22. The phase diagrams with respect to the external longitudinal random-field and the second-order phase transition temperature are calculated for the given values of the transverse field, z and p. It was also observed that the system exhibits reentrant phenomena which may be caused by the competition between the randomness of the external longitudinal magnetic field and the quantum effects. (c) 2005 Elsevier B.V. All rights reserved.