Select the most appropriate option to fill in the blank. After a lot of…

Concept

An idiom is a fixed multi-word expression whose meaning cannot be worked out from the ordinary meanings of its separate words. Because the expression is fixed, its verb slot is locked: swapping in a verb of similar meaning destroys the idiom, even when that substitute verb is perfectly sensible on its own.

Applying it here

The sentence says that people ended their differences after a long discussion, so the blank needs the verb that completes the established English idiom for making peace after a quarrel. That idiom is "bury the hatchet", which means to stop fighting and become friendly again. The image behind it is the old North American practice of literally burying weapons once hostilities were over, so the verb carries the sense of putting a conflict permanently out of sight; the phrase "end their differences" in the sentence restates exactly that meaning.

Word-by-word contrast

Compare how each offered verb behaves with the noun "the hatchet":

Only one verb produces the fixed idiom, so the completed sentence reads: "After a lot of discussion, they finally decided to bury the hatchet and end their differences."