We demonstrate that using up to seven stellar abundance ratios can place observational constraints on the star formation histories (SFHs) of Local Group dSphs, using Sculptor dSph as a test case. We use a one-zone chemical evolution model to fit the overall abundance patterns of {alpha} elements (which probe the core-collapse supernovae that occur shortly after star formation), s-process elements (which probe AGB nucleosynthesis at intermediate delay times), and iron-peak elements (which probe delayed Type Ia supernovae). Our best-fit model indicates that Sculptor dSph has an ancient SFH, consistent with previous estimates from deep photometry. However, we derive a total star formation duration of ~0.9Gyr, which is shorter than photometrically derived SFHs. We explore the effect of various model assumptions on our measurement and find that modifications to these assumptions still produce relatively short SFHs of duration <~1.4Gyr. Our model is also able to compare sets of predicted nucleosynthetic yields for supernovae and AGB stars, and can provide insight into the nucleosynthesis of individual elements in Sculptor dSph. We find that observed [Mn/Fe] and [Ni/Fe] trends are most consistent with sub-MCh Type Ia supernova models, and that a combination of "prompt" (delay times similar to core-collapse supernovae) and "delayed" (minimum delay times >~50Myr) r-process events may be required to reproduce observed [Ba/Mg] and [Eu/Mg] trends.