We report Chandra ACIS observations of the fields of 4 QSOs showing strong extended optical emission-line regions. Two of these show no evidence for significant extended X-ray emission. The remaining two fields, those of 3C 249.1 and 4C 37.43, show discrete (but resolved) X-ray sources at distances ranging from ~10 to ~40 kpc from the nucleus. In addition, 4C 37.43 also may show a region of diffuse X-ray emission extending out to ~65 kpc and centered on the QSO. It has been suggested that extended emission-line regions such as these may originate in the cooling of a hot intragroup medium. We do not detect a general extended medium in any of our fields, and the upper limits we can place on its presence indicate cooling times of at least a few 10^9 years. The discrete X-ray emission sources we detect cannot be explained as the X-ray jets frequently seen associated with radio-loud quasars, nor can they be due to electron scattering of nuclear emission. The most plausible explanation is that they result from high-speed shocks from galactic superwinds resulting either from a starburst in the QSO host galaxy or from the activation of the QSO itself. Evidence from densities and velocities found from studies of the extended optical emission around QSOs also supports this interpretation.
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