  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.
