Suppose $a, b, c \in I$ such that greatest common divisor of $x^2 + ax + b$ and $x^2 + bx + c$ is $(x + 1)$ and the least common multiple...

Question

Suppose $a, b, c \in I$ such that greatest common divisor of $x^2 + ax + b$ and $x^2 + bx + c$ is $(x + 1)$ and the least common multiple of $x^2 + ax + b$ and $x^2 + bx + c$ is $(x^3 - 4x^2 + x + 6)$. Find the value of $|a + b + c|$.

My attempt :
$$x^2 + ax + b = (x + 1)Y$$ $$x^2 + bx + c = (x + 1)Z$$

From here onwards I do not how to continue. Please help. Thank you!

Answer 1

We know that $x^2+ax+b$ and $x^2+bx+c$ have the same factor $x+1$.

But $$x^3-4x^2+x+6=x^3+x^2-5x^2-5x+6x+6=$$ $$=(x+1)(x^2-5x+6)=(x+1)(x-2)(x-3),$$ which gives that our polynomials they are: $$(x+1)(x-2)=x^2-x-2$$ and $$(x+1)(x-3)=x^2-2x-3.$$ Id est, $b=-2$, $a=-1$, $c=-3$ and $$|a+b+c|=6.$$

Answer 2

Using GCD $\times$ LCM = Product, we get $$(x+1)(x^3 - 4x^2 + x + 6) = (x^2+ax+b)(x^2+bx+c)$$ Now it remains to equate coefficients on both sides to get $(a, b, c) = (-1, -2, -3)$