  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 M32's mass, while 55%+/-21% of M32's 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 M32's
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 M31's disk at F2 (\sim1 disk scale length) to be 5-9 Gyr, our results
agree with an inside-out disk formation scenario for M31's disk.
