x4 - 1
------------------
 x3 + x2 + x + 1

Factor the numerator completely:

                x4 - 1

That's the difference of two perfect squares, so

             (x2 - 1)(x2 + 1)

The first factor is also the difference of two
perfect squares, so the complete factorization is:

         (x - 1)(x + 1)(x2 + 1)

(Note that you CANNOT factor the SUM of two perfect
 squares, only the DIFFERENCE of two perfect squares)

Factor the denominator completely, by grouping:

 x3 + x2 + x + 1

Factor x2 out of the first two terms:

 x2(x + 1) + x + 1

Factor 1 out of the last two terms:

 x2(x + 1) + 1(x + 1)

Now factor (x + 1) out of both terms:

 (x + 1)(x2 + 1)

Now return to the original expression:

      x4 - 1
------------------
 x3 + x2 + x + 1

and replace the numerator and denominator
by their complete factorizations:

  (x - 1)(x + 1)(x2 + 1)  
-------------------------
    (x + 1)(x2 + 1)

Cancel the (x + 1)'s and the (x2 + 1)'s

            1       1
  (x - 1)(x + 1)(x² + 1)  
-------------------------
    (x + 1)(x² + 1)
       1       1

and all you have left is a measly:

      x - 1
 
Edwin