Abstract: In a previous work we presented a theoretical analysis of the Lambda(b) -> J/psi K- p reaction based on which a recent experiment by the LHCb collaboration at CERN claimed the existence of two hidden charm pentaquarks, P-c(4380)(+) and P-c(4450)(+). In that work we focused only on the Lambda(1405) and P-c(4450)(+) signals and discussed the possible explanation of this pentaquark state within the picture of a dynamical meson-baryon molecule made up mostly from (D) over bar*Sigma(c) and (D) over bar*Sigma(c)* components. In the present work we improve upon the previous one by considering the total K- p and J/psi p data including all the relevant resonances contributing to the spectra, and discuss the possible nature of both P-c(4380)(+) and P-c(4450)(+). We also discuss several important topics, like the effect of the contact term in the reaction, the viability of reproducing the data without the P-c(4380)(+) and the possible quantum number assignment to these pentaquarks.