$$x^3-2x^2-5x+6$$
I want to get the solutions of this. I did a polynomial division.
First, I know that $(x+1)$ is a factor since $1^3-2\cdot1^2-5\cdot1+6 = 0$
So my division goes like this: $$..........x^2-3x-2$$ $$(x+1)|\overline{x^3-2x^2-5x+6}$$ $$x^3+x^2$$ $$......\overline{0-3x^2}-5x$$ $$........-3x^2-3x$$ $$...............\overline{0-2x}+6$$ $$....................2x-2$$ $$....................\overline{0 + 8}$$
(Sorry for the improvised formatting. Ignore any dots you see there.)
So I get, at the end, the quadratic $x^2-3x-2+8=x^2-3x+6$
However, $\triangle = (-3)^2-4(1)(6)=9-4(6)=9-24=-15$
Therefore, there are no solutions since $\triangle$ is negative.
But I definitely did something wrong, since I do know that the solutions are $1,-2,3$
What did I do wrong?