We use deep HST ACS/HRC observations of a field within M32 (F1) and an M31 background field (F2) to determine the star formation history (SFH) of M32 from its resolved stellar population. We find that 2-5Gyr old stars contribute som40%+/- 17% of M32s mass, while 55%+/-21% of M32s mass comes from stars older than 5 Gyr. The mass-weighted mean age and metallicity of M32 at F1 are <Age>=6.8+/-1.5 Gyr and <[M/H]>=-0.01+/-0.08 dex. The SFH additionally indicates the presence of young (<2 Gyr old), metal-poor ([M/H]sim-0.7) stars, suggesting that blue straggler stars contribute ~2% of the mass at F1; the remaining sim3% of the mass is in young metal-rich stars. Line-strength indices computed from the SFH imply a light-weighted mean age and metallicity of 4.9 Gyr and [M/H] = -0.12 dex, and single-stellar-population-equivalent parameters of 2.9+/-0.2 Gyr and [M/H]=0.02+/-0.01 dex at F1 (~2.7 re). This contradicts spectroscopic studies that show a steep age gradient from M32s center to 1re. The inferred SFH of the M31 background field F2 reveals that the majority of its stars are old, with sim95% of its mass already acquired 5-14 Gyr ago. It is composed of two dominant populations; sim30%+/-7.5% of its mass is in a 5-8 Gyr old population, and sim65%+/-9% of the mass is in a 8-14 Gyr old population. The mass-weighted mean age and metallicity of F2 are <Age>=9.2+/-1.2 Gyr and <[M/H]>=-0.10+/-0.10 dex, respectively. Our results suggest that the inner disk and spheroid populations of M31 are indistinguishable from those of the outer disk and spheroid. Assuming the mean age of M31s disk at F2 (sim1 disk scale length) to be 5-9 Gyr, our results agree with an inside-out disk formation scenario for M31s disk.
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