SOLUTION: How do you simplify by dividing n^3-27/n-3

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This falls under the "special factoring" case -- "difference of cubes"
a^3 – b^3 = (a – b)(a^2 + ab + b^2)
For more info see:
http://www.purplemath.com/modules/specfact2.htm
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Therefore for,
n^3-27/n-3
we can rewrite as:
(n^3 - 3^3)/n-3
the numberator can be then be factored:
(n-3)(n^2 + 3n + 9)/(n-3)
the (n-3) cancels leaving:
n^2 + 3n + 9