We analyze the decay B0→KS0π+π- using a sample of 232×106 Υ(4S)→BB̄ decays collected with the BABAR detector at the SLAC PEP-II asymmetric-energy B factory. A maximum likelihood fit finds the following branching fractions: B(B0→K0π+π-)=(43.0±2.3±2.3) ×10-6,B(B0→f0(→π+π-)K0)=(5.5±0.7±0. 5±0.3)×10-6 and B(B0→K*+π-)=(11.0±1. 5±0.5±0.5)×10-6. For these results, the first uncertainty is statistical, the second is systematic, and the third (if present) is due to the effect of interference from other resonances. We also measure the CP-violating charge asymmetry in the decay B0→K*+π-, AK*π=-0. 11±0.14±0.05. © 2006 The American Physical Society.