We present the results of X-ray imaging and spectroscopic analysis of a sample of Seyfert 2 galaxies that contain starbursts, based on their optical and UV characteristics. These composite galaxies exhibit extended, soft, thermal X-ray emission, which we attribute to their starburst components. Comparing their X-ray and far-infrared properties with ordinary Seyfert and starburst galaxies, we identify the spectral characteristics of their various intrinsic emission sources. The observed far-infrared emission of the composite galaxies may be associated almost exclusively with star formation, rather than the active nucleus. The ratio of the hard X-ray luminosity to the far-infrared and [O III] 5007 luminosity distinguishes most of these composite galaxies from ``pure Seyfert 2 galaxies, while their total observed hard X-ray luminosity distinguishes them from ``pure starbursts. The hard nuclear X-ray source is generally heavily absorbed (N_H > 10^{23} cm^{-2}) in the composite galaxies. Based on these results, we suggest that the interstellar medium of the nuclear starburst is a significant source of absorption. The majority of the sample are located in groups or are interacting with other galaxies, which may trigger the starburst or allow rapid mass infall to the central black hole, or both. We conclude that starbursts are energetically important in a significant fraction of active galaxies, and starbursts and active galactic nuclei may be part of a common evolutionary sequence.
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