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Ultrashort, high-brightness X-ray pulses are an important tool in many areas of research. In principle, a laser-pumped free-electron laser (FEL) can generate coherent, polarized X-ray for this purpose. In a laser-pumped X-ray FEL an intense laser field replaces the magnetic wiggler field of a conventional FEL. Depending on the intensity and quality of both the electron beam and pump laser, the spontaneous incoherent X-ray, generated by Thomson backscattering, can be coherently amplified. In a conventional FEL configuration the generation of X-ray typically requires electron beam energies in the multi-GeV range. On the other hand, in a laser-pumped X-ray FEL, electron beam energies in the multi-MeV range would be sufficient. A laser-pumped X-ray FEL could in principle be a relatively compact source of coherent X-ray. However, to realize this concept a number of physics and technology issues must be resolved. Foremost among these are the stringent requirements placed on the electron beam quality and brightness as well as the pump laser intensity and energy. The seed radiation to the laser-pumped FEL is the laser induced spontaneous radiation. The gain length associated with the coherent X-ray as a function of energy spread on the thermal electron beam is analyzed. These issues, as well as others, necessary to achieve coherent amplified spontaneous X-ray generation in a laser-pumped FEL will be addressed.