Which of the townshend acts forced american colonists to allow british troops to stay in their homes?

Quartering Act is the correct answer.

Explanation:

The acts passed by the British parliament for local governments of American Colonies to provide food and housing to the British soldiers were known as Quartering Acts. The acts were amendments to the Mutiny Act and required yearly renewal. It was a source of contention between the colonies and Britain, but didn't met with significant resistance as the British troops were present if few colonies in significant numbers. It didn't require to house the soldiers in private homes but required the colonial governments to feed and shelter the troops sheltered in their colony.

d. migration of french settlers into english colonies