  We report a study of the decays B0 -> J/psi K(*)0 pi+ pi-, which involve the
creation of a u u-bar or d d-bar quark pair in addition to a b-bar -> c-bar(c
s-bar) decay. The data sample consists of 110 1/pb of p p-bar collisions at
sqrt{s} = 1.8 TeV collected by the CDF detector at the Fermilab Tevatron
collider during 1992-1995. We measure the branching ratios to be BR(B0 -> J/psi
K*0 pi+ pi-) = (8.0 +- 2.2 +- 1.5) * 10^{-4} and BR(B0 -> J/psi K0 pi+ pi-) =
(1.1 +- 0.4 +- 0.2) * 10^{-3}. Contributions to these decays are seen from
psi(2S) K(*)0, J/psi K0 rho0, J/psi K*+ pi-, and J/psi K1(1270).
