When there isn't a greatest common factor (gcf) to factor out, we move on to factoring the trinomial. 

     n² - 14n - 72 = (          )(          )

The terms that go on the left hand side of each set of parentheses are the pair of factors of the first term in the trinomial. The only pair of factors of n² are n and n, since  n·n = n².

The terms that go on the right hand side of each set of parentheses are the pair of factors of the last term in the trinomial. There are a few pairs of factors of -72, which are as follows:

          1 and -72 ;  -1 and 72

          2 and -36 ;  -2 and 36

          3 and -24 ;  -3 and 24

          4 and -18 ;  -4 and 18

          6 and -12 ;  -6 and 12

          8 and -9  ;  -9 and 8 

We need to pick the pair that will add up to equal the middle term in the trinomial. The only pair that adds up the -14 is 4 and -18. And so,

     n² - 14n - 72 = (n + 4)(n - 18)