A kinetically complete measurement of the C12(Be10, α+α+n) and (Be10, α+α) reactions has been performed at a beam energy of 30 MeV/nucleon. The charged beam velocity particles were detected in an array of Si-CsI detectors placed at zero degrees, and the neutrons in an 81-element neutron array. The coincident detection of the final-state particles, produced in the breakup of Be10, allowed the reconstruction of the excitation energy in the Be8 and Be9 systems. States in Be8 were identified, in particular the ground and first-excited states; and in Be9, states at 1.68, 2.43, and (2.78, 3.05) MeV were observed. The population of these levels, in particular the 2.43 MeV 5/2- level, suggests that collective excitations play an important role in the neutron removal process. Distorted wave Born approximation and Glauber-type calculations have been used to model the direct neutron removal from the Be10 ground state and the two-step removal via inelastic excitations of the Be10(2+) and Be9(5/2-) excited states.