The primary motivations for British expansion into the Ohio Valley in the mid-1700s were:
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To gain access to the region's natural resources: The Ohio Valley was rich in resources such as fertile land, timber, and potential for trade, which were attractive to British settlers and investors.
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To force the French out of North America: The British sought to diminish French influence and presence in the region, as the French were already established in parts of North America, including Canada and the Ohio Valley.
These factors were closely tied to the broader struggle for power between Britain and France in North America, which eventually culminated in the French and Indian War (1754-1763).
While demonstrating British strength to France and Spain and putting French settlers under English rule were also considerations, the key drivers were access to natural resources and territorial control to reduce French influence. Thus, the most accurate reasons would be mainly the first two points listed.