Many potential applications of semiconductor nanocrystals are hindered by nonradiative Auger recombination wherein the electron-hole (exciton) recombination energy is transferred to a third charge carrier. This process severely limits the lifetime and bandwidth of optical gain, leads to large nonradiative losses in light-emitting diodes and photovoltaic cells, and is believed to be responsible for intermittency ("blinking") of emission from single nanocrystals. The development of nanostructures in which Auger recombination is suppressed has recently been the subject of much research in the colloidal nanocrystal field. Here, we provide direct experimental evidence that so-called "giant" nanocrystals consisting of a small CdSe core and a thick CdS shell exhibit a significant (orders of magnitude) suppression of Auger decay rates. As a consequence, even multiexcitons of a very high order exhibit significant emission efficiencies, which allows us to demonstrate optical amplification with an extraordinarily large bandwidth (>500 meV) and record low excitation thresholds. This demonstration represents an important milestone toward practical lasing technologies utilizing solution-processable colloidal nanoparticles.