How do you simplify x^2-4x-21 divided by x+3

We can factor x^2-4x-21 to get (x+3)(x-7)

Notice how 
3 plus -7 = -4
3 times -7 = -21
which helps us factor the trinomial

After you've factored it, we can cancel out a pair of (x+3) terms leaving us with the final answer of x-7

Note: the cancellation happens because (x+3)/(x+3) = 1. Divide anything (except 0) over itself and you'll get 1. A simpler example of this is 5/5 = 1.