3x⁴ = 3x³ • x, and
3x³ (x - 2) = 3x⁴ - 6x³
Subtract this from the numerator to get a remainder of
(3x⁴ - 8x³ - x² + 9x + 5) - (3x⁴ - 6x³) = -2x³ - x² + 9x + 5
-2x³ = -2x² • x, and
-2x² (x - 2) = -2x³ + 4x²
Subtract this from the previous remainder to get a new remainder of
(-2x³ - x² + 9x + 5) - (-2x³ + 4x²) = -5x² + 9x + 5
-5x² = -5x • x, and
-5x (x - 2) = -5x² + 10x
Subtract this from the last remainder to get a new one of
(-5x² + 9x + 5) - (-5x² + 10x) = -x + 5
-x = -1 • x, and
-1 (x - 2) = -x + 2
This gives a new remainder of
(-x + 5) - (-x + 2) = 3
3 does not divide x, so we're done.
So, we have
(3x⁴ - 8x³ - x² + 9x + 5) / (x - 2) = 3x³ - 2x² - 5x - 1 + 3/(x - 2)