If x³ - 6x² + ax + b is divisible by (x² - 3x + 2), then the values of a and b are:

x² - 3x + 2 = 0
⇒ x² - 2x - x + 2 = 0
⇒ x(x - 2) - 1(x - 2) = 0
⇒ (x - 2)(x - 1) = 0
x = 2 and 1
When x = 2, then
x³ - 6x² + ax + b = 0
⇒ 8 - 24 + 2a + b = 0
⇒ 2a + b = 16 . . . . . . (i)
When x = 1, then
x³ - 6x² + ax + b = 0
⇒ 1 - 6 + a + b = 0
⇒ a + b = 5 . . . . . . (ii)
From equation (ii)
a + 5 = 16
a = 11, b = -6