I fixed your Latex code. Take a look so you know how to write fractions now
Oh no! You changed it back to wrong.... sigh. When writing fractions in Latex, use \frac{numerator}{denominator}
I spotted your error. You don't need to plug in s=16 and get 0 in the numerator. 16-s and s-16 are only a negative sign factored out away from equal.
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Now the 16-s cancels and you are left with
I wouldn't call it "tossing in a negative".
Here, think about it. Let's say you have (a-b). Now if you factor out a -1 from both terms, what do you get? -1(-a+b) or -1(b-a) or -(b-a). If you distribute the -1 back through both terms you get your original sum.