(x-1)(x-2)+(x-2)x+x(x-1) = 3(x-α)(x-β)
に
x = 0 , 1 , 2
を代入すると

(0-1)(0-2)+(0-2)0+0(0-1) = 3(0-α)(0-β)
2 = 3αβ
αβ = 2/3

(1-1)(1-2)+(1-2)1+1(1-1) = 3(1-α)(1-β)
-1 = 3(α-1)(β-1)
(α-1)(β-1) = -1/3

(2-1)(2-2)+(2-2)2+2(2-1) = 3(2-α)(2-β)
2 = 3(α-2)(β-2)
(α-2)(β-2) = 2/3

後は代入するだけ