Final answer:
Herbert Hoover's greatest mistake in dealing with the Bonus Army was ordering the military to forcibly disperse the veterans with tear gas and bayonets, leading to violence and public outrage.
Step-by-step explanation:
Herbert Hoover's greatest mistake in his handling of the Bonus Army was ordering the military to confront the veterans with tear gas and bayonets, which resulted in panic, riots, and casualties, including the death of an infant. This action took place on July 28, 1932, when Hoover sought to disperse the World War I veterans who had camped in Washington, D.C., to demand early payment of their bonuses during the severe economic downturn of the Great Depression. The violent eviction of the Bonus Army and the burning of their camp by U.S. troops marked a significant blow to Hoover's presidency and played a role in his subsequent loss in the 1932 presidential election to Franklin Roosevelt.