Answer
See below.
Work Step by Step
His test scores average to $(86+80+84+90)/4=85$
Thus if his final exam's score is $x$, then his average will be $2x/3+85/3$
If we want an average of at least $80$: $80\leq2x/3+85/3\\155/3\leq2x/3\\155\leq2x\\77.5\leq x$
If we want an average of at least $90$: $90\leq2x/3+85/3\\185/3\leq2x/3\\185\leq2x\\92.5\leq x$
