Rewrite this sentence to make it correct: the boston massacre and the repeal of taxes under the townshend acts began huge protests across the colonies.

There are many ways in which you can rewrite this sentence in order to make it correct, but the bottom line is that the verb began is incorrect here.
So, here are some examples:
The Boston Massacre and the repeal of taxes under the Townshend Acts resulted in/led to/provoked/gave rise to huge protests across the colonies.