We report a study of the decays B0→J/ψK(*)0π+π−, which involve the creation of a uu¯ or dd¯ quark pair in addition to a b¯→c¯(cs¯) decay. The data sample consists of 110pb−1 of pp¯ collisions at s√=1.8TeV collected by the CDF detector at the Fermilab Tevatron collider during 1992–1995. We measure the branching fractions to be B(B0→J/ψK*0π+π−)=(6.6±1.9±1.1)×10−4 and B(B0→J/ψK0π+π−)=(10.3±3.3±1.5)×10−4. Evidence is seen for contributions from ψ(2S)K(*)0, J/ψK0ρ0, J/ψK*+π−, and J/ψK1(1270).