The expansion of slavery is the correct answer because during these years the northern states and southern states were heavily divided on the issue. The northern states mostly believed in not allowing slavery to expand to the territories, while the southern states believed that the territories should be allowed to decide for themselves. Though the other choices did cause some controversy they were not as politically divisive as the issue of slavery.

The Louisiana Purchase doubled the United States in size and allowed it to expand westward. Americans took advantage of the new, cheap land that was just bought by moving westward. The Mexican-American War continued the expansion of the United States into Texas and New Mexico, but it did not start the westward expansion. The Homestead Act encouraged westward expansion by giving land to Union soldiers who fought in the Civil War, but it did not begin the expansion. The Transcontinental Railroad accelerated expansion by making travel easier, but it did not begin the expansion.

Polk negotiated the Oregon territory from Great Britain, annexed Texas, and started the Mexican-American war to gain most of the modern Southwest United States

The Frontier Thesis was the idea that American Democracy was innately tied to the American Frontier, and that the frontier needed to continue expanding for democracy to flourish. Settlers on the frontier were able to experiment with new ideas on how best to establish towns and local government. Historically, the western frontier was traditionally more liberal than conservative, older states back east. This propelled the advancement and consistent innovation of American Democracy. Frederick Jackson Turner presented the idea in 1893 at a conference in Chicago. At the time, the U.S. land frontier to the Pacific had largely faded as California was admitted to the union in 1850. The Philippines became a foreign U.S. possession in 1898 and many politicians thereafter insisted on its admittance to the union, including Theodore Roosevelt. 

Manifest Destiny was the belief that American settlers were destined to conquer the North American mainland through the exceptionalism of the American people and its unique democratic institutions. In Gast's painting, Lady Liberty is marching along with settlers from the bright and enlightened east to the dark and unknown west. Behind her, trains forge west while in front of her, "savage" Indians are running further into the darkness, away from the settlers. Further indications of the time frame include the covered wagons making the journey west and the farming techniques that were well-implemented in the east in the foreground of the painting. 

Image is in the public domain, accessed through Wikipedia Media Commons: https://commons.wikimedia.org/wiki/File:American_progress.JPG

Constantly at war with his European neighbors, Napoleon's coffers emptied quickly. In order to replenish his financial stores, Napoleon looked to sell land in the Americas, finding a very willing buyer in Thomas Jefferson. With the Louisiana Purchase, Napoleon was able to soldier on while the United States added a huge swath of new territory.

In 1867, Secretary of State William Seward purchased Alaska from Russia for a paltry 7.2 million dollars. Although some criticized the purchase at the time, referring to it "Seward's Folly," reaction to the move was generally positive. The move had the effect of weakening both English and Russia colonies in the region, and the natural resources found in Alaska have benefited the country greatly.

Oregon Country referred to the territory that encompasses nearly all of the Pacific Northwest, including British Columbia, Idaho, Oregon, Washington, and parts of Wyoming and Montana. Those in favor of the slogan hoped to gain all of the region for the United States. The border was set, however, at the 49th parallel.