Hmm. Yes 116 is actually correct - my mistake.
What I said would only apply if the velocity was a linear equation, e.g. v = 2 + 4t or something like that.
Since your velocity equation ended up being non linear (a quadratic), you'd have to use the fact that average velocity = total displacement / total time.
So in other words, from your new velocity equation, integrate it one more time to find the equation for displacement, then work out the total displacement between t = 5 and t = 8. (plug them into the displacement equation then minus from each other). Once you've got the total displacement between the two times, divide by total time taken (8 - 5 = 3 seconds). You should end up with 116.