We present long-slit spectroscopy of the elliptical galaxy M32, obtained with the 8 m Subaru telescope at Mauna Kea, the 1.5 m Tillinghast telescope at the F. L. Whipple Observatory, and the 4 m Mayall telescope at the Kitt Peak National Observatory. The spectra cover the Lick index red spectral region, as well as higher order Balmer lines in the blue. Spectra have been taken with the slit offset from the nucleus to avoid scattered-light contamination from the bright nucleus of M32. An analysis of numerous absorption features, particularly involving the Hγ and Hβ Balmer lines, reveals that systematic radial trends are evident in the integrated spectrum of M32. Population synthesis models indicate a radial change in both the age and chemical composition of the light-weighted mean stellar population in M32, from the nucleus out to 33", i.e., an approximately 1.0 effective radius, Re. Specifically, the light-weighted mean stellar population at 1Re is older by ~3 Gyr and more metal-poor by about -0.25 dex in [Fe/H] than the central value of ~4 Gyr and [Fe/H]~0.0. We show that this apparent population trend cannot be attributed to a varying contribution from either hot stars or emission-line contamination. The increase in age and decrease in metal abundance with radius are sufficiently well matched to explain the flat radial color profiles previously observed in M32. In addition, the ratio of Mg to Fe abundance, [Mg/Fe], increases from about -0.25 in the nucleus to about -0.08 at 1Re. Finally, we find spuriously pronounced line-strength gradients in the Mayall data that are an artifact of scattered light from the bright nucleus. Scattered-light issues may explain the lack of consistency among previously published studies of radial line strength gradients in M32.