To make busy; to occupy or keep busy with activity.
From the prefix 'be-' plus 'busy.' This creates a verb meaning to cause someone to be busy, following the intensive or causative function of the 'be-' prefix.
Medieval and Early Modern writers created 'bebusy' to mean 'to occupy thoroughly' or 'to keep constantly busy,' and this same 'be-' prefix also created 'becalm' (to make calm) and 'befuddle' (to confuse)—it's one of English's most flexible verb-making tools.
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