Final answer:
President Polk aimed to expand the U.S. territory, trying to buy California and New Mexico from Mexico, but resorted to provoking a war when negotiations failed, leading to the Mexican American War and territorial gains for the U.S.
Step-by-step explanation:
President James K. Polk's expansionist vision for the United States included the acquisition of new territories, notably those held by Mexico. During his administration in the 1840s, the American public and Polk were proponents of Manifest Destiny, believing it was the United States' divine right to expand across the continent. Polk tried to purchase Upper California and New Mexico from Mexico, offering up to $25 million for California alone. However, when diplomatic efforts failed, and Mexico refused to sell these territories, Polk saw an opportunity to provoke war after a skirmish at the disputed Texas border.
War with Mexico would solidify U.S. claims to Texas and potentially allow for the annexation of further territories. This war, ultimately called the Mexican-American War, resulted in significant territorial gains for the United States, with the cession of California and New Mexico through the Treaty of Guadalupe Hidalgo in 1848. Though this fulfilled expansionist desires, it was also criticized as an aggressive move to extend slavery and increase slaveholding territory, leading to deepening sectional conflicts within the U.S.