Find the quotient and remainder using synthetic division

x^5 - x^4 + 5x^3 - 5x^2 +1x - 2/ x-1

= (x⁵ - x⁴ + x³ - 5x² + x - 2)/(x - 1)
= x⁴ + ([x⁵ - x⁴ + x³ - 5x² + x - 2] - [x⁵ - x⁴])/(x - 1)
= x⁴ + (x³ - 5x² + x - 2)/(x - 1)
= x⁴ + x² + ([x³ - 5x² + x - 2] - [x³ - x²])/(x - 1)
= x⁴ + x² + (- 3x² + x - 2)/(x - 1)
= x⁴ + x² - 3x + ([- 3x² + x - 2] - [- 3x² + 3x])/(x - 1)
= x⁴ + x² - 3x + (- 2x - 2)/(x - 1)
= x⁴ + x² - 3x - 2 + ([- 2x - 2] - [- 2x - 2])/(x - 1)
= x⁴ + x² - 3x - 2 remainder 0

Answer: x⁴ + x² - 3x - 2 (no remainder)