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We report a study of the decaysB0 →J/ψK(*)0π+π-, which involve the creation of a uu̅ or dd̅ quark pair in addition to a b̅→ c̅(cs̅) decay. The data sample consists of 110 pb-1 of pp̅ collisions at √s=1.8 TeV collected by the CDF detector at the Fermilab Tevatron collider during 1992–1995. We measure the branching fractions to be Ɓ(B0 → J/ψK(*)0π+π-) = (6.6 ± 1.9 ± 1.1) x 10-4 and Ɓ(B0 → J/ψK(*)0π+π-)=(10.3 ± 3.3 ± 1.5) x 10-4 . Evidence is seen for contributions from ψ (2S) K(*)0, J/ψK0p0, J/ψK*+ π-, and J/ψK1(1270).