# text{ -------------------------} #
# x -1 quad text{)} quad x^3 + 3x^2 - 3x - 2 #
That's a pain to format. Anyway, the first "digit", first term in the quotient, is #x^2#. We compute the digit times #x-1#, and take that away from #x^3 + 3x^2 - 3x -2 #:
#text{ } x^2#
# text{ -------------------------} #
# x -1 quad text{)} quad x^3 + 3x^2 - 3x - 2 #
# text{ } x^3 -x^2 #
# text{ ----------------} #
# text{ } 4 x^2 - 3x - 2#
OK, back to the quotient. The next term is #4x# because that times #x# gives #4 x^2#. After that the term is #1#.
#text{ } x^2 + 4 x + 1#
# text{ ------------------------- #
# x -1 quad text{)} quad x^3 + 3x^2 - 3x - 2 #
# text{ } x^3 -x^2 #
# text{ ----------------} #
# text{ } 4 x^2 - 3x - 2#
# text{ } 4 x^2 - 4x#
# text{ ----------------} #
# text{ } x - 2 #
# text{ } x - 1 #
# text{ --------} #
# text{ } -1#
We have a non-zero remainder! That says
# x^3 + 3x^2 - 3x - 2 = ( x -1)(x^2 + 4x + 1) - 1 #
