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

Let us consider, 

p(x) = x³ – 3x² + x + 2, q(x) = x – 2 and  r (x) = –2x + 4 
By Division Algorithm, we know that: p(x) = q(x) × g(x) + r(x) 
Therefore, 
x³ – 3x² + x + 2 = (x – 2) × g(x) + (–2x + 4) 
⇒ x³ – 3x² + x + 2 + 2x – 4 = (x – 2) × g(x) 
⇒ g(x) = $\frac{({\mathrm{x}}^3-3{\mathrm{x}}^2+3\mathrm{x}-2)}{(\mathrm{x} - 2)}$ 
On dividing  x³ – 3x² + x + 2  by x – 2,we get 
⇒ g(x) = x² – x - 1.