In a laser-pumped x-ray free-electron laser (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 Thomson backscattered radiation can be coherently amplified. In a conventional FEL the generation of x rays requires electron beam energies in the multi-GeV range. In a laser-pumped x-ray FEL, electron beam energies in the multi-MeV range would be sufficient. To generate coherent x rays with this mechanism a number of physics and technology issues must be addressed. Foremost among these are the stringent requirements placed on the electron beam quality and brightness as well as on the pump laser. The seed radiation for the laser-pumped FEL is the laser-induced spontaneous radiation. The evolution of incoherent radiation into coherent radiation as well as the power gain lengths associated with the coherent x rays are analyzed and discussed. There is excellent agreement between our analytical results and GENESIS simulations for the radiated power, gain length, conversion efficiency, linewidth, and saturation length. These issues, as well as others, necessary to achieve coherent amplified x rays in a laser-pumped FEL are discussed. While a coherent x-ray source would have a number of attractive features, the requirements placed on both the electron beam and pump laser are extremely challenging.