Final answer:
The slogan 'Fifty-four forty or fight!' was related to the American expansionist demand to control the entire Oregon Territory against British claims, which was eventually resolved with the Oregon boundary settlement at the 49th parallel.
Step-by-step explanation:
The Disputed Oregon Territory
The slogan "Fifty-four forty or fight!" was a call for American expansion and represented a demand for U.S. control over the entire Oregon Territory, which extended up to the latitude of 54° 40'. This slogan became associated with the presidential campaign of James K. Polk in 1844. The dispute over this territory was not with Mexico, as Mexico was contending with the U.S. over the Texas region, but rather with Great Britain. The Oregon Territory dispute between the U.S. and Great Britain was initially based on claims by both nations to land in the Pacific Northwest.
Ultimately, President Polk, while initially supporting the expansionist slogan, was willing to compromise for a boundary at the 49th parallel, which is north of the Columbia River and which today forms part of the border between the United States and Canada. The British agreed to the proposal in June 1846, which allowed both nations to avoid military conflict over the region, and secured the division of Oregon Territory at the 49th parallel, with Great Britain retaining land to the north and the United States acquiring land to the south, including the strategic deepwater port of Puget Sound.