We report a study of the decays B0 --> J/psiK(*)0pi(+)pi(-), which involve the creation of a uu or dd quark pair in addition to a b -->c(cs) decay. The data sample consists of 110 pb(-1) of pp collisions at square root[s] = 1.8 TeV collected by the CDF detector at the Fermilab Tevatron collider during 1992-1995. We measure the branching fractions to be B(B0 --> J/psiK(*0)pi(+)pi(-)) = (6.6 +/- 1.9 +/- 1.1)x10(-4) and B(B0 --> J/psiK0pi(+)pi(-)) = (10.3 +/- 3.3 +/- 1.5)x10(-4). Evidence is seen for contributions from psi(2S)K(*)0, J/psiK0rho(0), J/psiK(*+)pi(-), and J/psiK1(1270).
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