Don't know where you get your working after "Critical numbers". I make it the same result after differentiating, but all you need to do is note that when f'(t)=0 you're just interested in the top, and 64(8+t)^2 is a common factor of both terms. You're left with:
8+t = 3t
(unless I've made a mistake)
and it should be fairly straightforward from there.