polynomial Question !

Let P(x) = Q(x)(x - 2)(x - 3) + Ax + B
Remainder theorem
P(2) = Q(2)(2 - 2)(2 - 3) + 2A + B = 3 => 2A + B = 3...(1)
P(3) = Q(3)(2 - 3)(3 - 3) + 3A + B = 2 => 3A + B = 2...(2)
(2) - (1) => A = -1
Sub into (1), 2(-1) + B = 3 => B = 5
Therefore P(x) = Q(x)(x - 2)(x - 3) - x + 5
The remainder when P(x) is divided by (x - 2)(x - 3) is - x + 5