Why did the American colonists want to expand control into the Ohio Valley in the mid-1700s
to gain access to the region's natural resources
to demonstrate Britain's strength to France and Spain
to force the French out of North America
to put French settlers there under English rule
which is right?
The American colonists' desire to expand into the Ohio Valley during the mid-1700s was driven by a few reasons, among which gaining access to the region's natural resources was a significant factor. The Ohio Valley was rich in resources such as fur-bearing animals, fertile land for farming, and later, was understood to be abundant in minerals.
However, from a geopolitical perspective, the expansion into the Ohio Valley was also part of a larger struggle for control of North America between Britain and France, which culminated in the French and Indian War (known as the Seven Years' War globally). The British wanted to force the French out of North America and demonstrate their strength to both France and Spain, who were European rivals with colonial interests in the New World.
The desire to put French settlers under English rule might have been a subsequent consequence of the British victory in the region but was not a primary motivation for expanding into the Ohio Valley.
Given these considerations, the statement that most broadly encapsulates the motivations is: "to force the French out of North America." This goal encompasses the strategic motivation behind the British and colonists' expansion into the region, which was securing it from French control and thereby consolidating British influence in North America.