This paper outlines our studies of bifurcation,quasi-periodic road to chaos and other dynamic characteristics in an exteal-cavity multi-quantum-well laser with delay optical feedback.The bistable state of the laser is predicted by finding theoretically that the gain shifts abruptly between two values due to the feedback.We make a linear stability analysis of the dynamic behavior of the laser.We predict the stability scenario by using the characteristic equation while we make an approximate analysis of the stability of the equilibrium point and discuss the quantitative criteria of bifurcation.We deduce a formula for the relaxation oscillation frequency and prove theoretically that this formula function relates to the loss of carriers transferring between well regime and barrier regime,the feedback level,the delayed time and the other intrinsic parameters.We demonstrate the dynamic distribution and double relaxation oscillation frequency abruptly changing in periodic states and find the multi-frequency characteristic in a chaotic state.We illustrate a road to chaos from a stable state to quasi-periodic states by increasing the feedback level.The effects of the transfers of carriers and the escaping of carriers on dynamic behavior are analyzed,showing that they are contrary to each other via the bifurcation diagram.Also,we show another road to chaos after bifurcation through changing the linewidth enhancement factor,the photon loss rate and the transfer rate of carriers.