Measurements of neutral<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi></mml:math>decay branching fractions to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msubsup><mml:mi>K</mml:mi><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msubsup><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math>final states and the charge asymmetry of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mrow><mml:mo>*</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math>

We analyze the decay ${B}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ using a sample of $232\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $\ensuremath{\Upsilon}(4S)\ensuremath{\rightarrow}B\overline{B}$ decays collected with the BABAR detector at the SLAC PEP-II asymmetric-energy $B$ factory. A maximum likelihood fit finds the following branching fractions: $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{K}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})=(43.0\ifmmode\pm\else\textpm\fi{}2.3\ifmmode\pm\else\textpm\fi{}2.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6},\text{ }\text{ }\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{f}_{0}(\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}){K}^{0})=(5.5\ifmmode\pm\else\textpm\fi{}0.7\ifmmode\pm\else\textpm\fi{}0.5\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{K}^{*+}{\ensuremath{\pi}}^{\ensuremath{-}})=(11.0\ifmmode\pm\else\textpm\fi{}1.5\ifmmode\pm\else\textpm\fi{}0.5\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$. 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 ${B}^{0}\ensuremath{\rightarrow}{K}^{*+}{\ensuremath{\pi}}^{\ensuremath{-}}$, ${\mathcal{A}}_{{K}^{*}\ensuremath{\pi}}=\ensuremath{-}0.11\ifmmode\pm\else\textpm\fi{}0.14\ifmmode\pm\else\textpm\fi{}0.05$.

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