(x^3+x^2-18x+2)/(x+5)

This works just like "long  division".......we have

                      x^2   - 4x  + 2  

         ----------------------------

x+5  ) x^3 +   x^2 - 18x +  2

          x^3  + 5x^2

         ----------------------------

                   -4x^2  - 18x

                   -4x^2  - 20x

                   ---------------

                                 2x  +  2

                                 2x  + 10

                                 -----------

                                          -8

So, there's our answer......  x^2 - 4x  + 2  with a "remainder" of -8 /(x+5)

Note that if we can factor the numerator/denominator, this sometimes "cancels" terms and division might not be necessary at all, but that wasn't possible here.