We report molecular gas observations of IRAS 20100 - 4156 and IRAS 03158 + 4227 , two local ultraluminous infrared galaxies ( ULIRGs ) hosting some of the fastest and most massive molecular outflows known . Using ALMA and PdBI observations , we spatially resolve the CO ( 1 - 0 ) emission from the outflowing molecular gas in both and find maximum outflow velocities of v _ { max } \sim 1600 and \sim 1700 km s ^ { -1 } for IRAS 20100 - 4156 and IRAS 03158 + 4227 , respectively . We find total gas mass outflow rates of \dot { M } _ { OF } \sim 670 and \sim 350 M _ { \odot } yr ^ { -1 } , respectively , corresponding to molecular gas depletion timescales \tau ^ { dep } _ { OF } \sim 11 and \sim 16 Myr . This is nearly 3 times shorter than the depletion timescales implied by star formation , \tau ^ { dep } _ { SFR } \sim 33 and \sim 46 Myr , respectively . To determine the outflow driving mechanism , we compare the starburst ( L _ { * } ) and AGN ( L _ { AGN } ) luminosities to the outflowing energy and momentum fluxes , using mid-infrared spectral decomposition to discern L _ { AGN } . Comparison to other molecular outflows in ULIRGs reveals that outflow properties correlate similarly with L _ { * } and L _ { IR } as with L _ { AGN } , indicating that AGN luminosity alone may not be a good tracer of feedback strength and that a combination of AGN and starburst activity may be driving the most powerful molecular outflows . We also detect the OH 1.667 GHz maser line from both sources and demonstrate its utility in detecting molecular outflows .