Final answer:
The United States acquired more territory through the 1803 Louisiana Purchase from France and the 1848 Treaty of Guadalupe Hidalgo after war with Mexico, both symbolic of the doctrine of Manifest Destiny.
Step-by-step explanation:
The two ways that the United States acquired more territory were through the purchase of the Louisiana Territory from France and through war with Mexico, leading to Mexico ceding land including Texas and other territories to the United States. The Louisiana Purchase, made in 1803 for $15 million, was a significant event under President Thomas Jefferson's administration, effectively doubling the size of the United States. Additionally, following the occupation of Mexico City, the Treaty of Guadalupe Hidalgo was signed in 1848, whereby Mexico ceded territories now known as California, Arizona, New Mexico, Nevada, Colorado, Utah, and part of Wyoming to the United States. The annexation of these lands was influenced by the doctrine known as Manifest Destiny, the belief that the U.S. was destined to expand across the continent.