ISL 1970 Problem 47

Given a polynomial $$P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),$$ where $a, b, c \neq 0$ , prove that $P(x)$ is divisible by $$Q(x) = abx^2 + (a^2 + b^2)x + ab$$ and conclude that $P(x_0)$ is divisible by $(a + b)^3$ for $x_0 = (a + b + 1)^n, n \in \mathbb N$ .