to finish off, To complete or conclude something, often with a final touch.
to finish up, To complete something, often implying the last steps or finalization.
Word Origin
Language
Old English
Original word
fīnan
Originally meant
to end, to complete
Background
The word 'finish' comes from the Old English 'fīnan', which means to end or complete. It has been used in English since the Middle Ages to describe the act of bringing something to an end.