On dividing x³ - 3x² + x + 2 by a polynomial g(x), the quotient and remainder were (x - 2) and (-2x + 4), respectively. Find g(x).

We know the formula,
Dividend = Divisor × quotient + Remainder
p(x) = g(x) × q(x) + r(x)
Substitute the values, we get
x³ - 3x² + x + 2 = g(x) × (x - 2) - 2x + 4
Add 2x and subtract 4 both side,
x³ - 3x² + x + 2 + 2x - 4 = g(x) × (x - 2)
Simplify and divide by x - 2
x³ - 3x² + 3x - 2 / x - 2 = g(x)
g(x) = (x - 2)(x² - x + 1) / x - 2
So, g(x) = x² - x + 1