How do I solve the equation $(x - 1)(x - 2)(x - 3) = (x - 2)(x - 3)(x - 4)$? Problem: $(x - 1)(x - 2)(x - 3) = (x - 2)(x - 3)(x - 4)$
Heres my question with this problem: why do I end up with a wrong answer when I divide both sides by $(x-2)(x-3)$ to cancel out the $(x-2)(x-3)$ on both sides. Is this not allowed and why? Please provide the explanation to this question. You do not have to solve the problem.
Thank you very much.

Obviously, both $x=2$ and $x=3$ are solutions.
If $x$ is neither above, then we can cancel $(x-2)(x-3)$ on both sides of the equation, so we get
$$
x-1 = x-4, \text{or} -1=-3, 
$$
which is a contradiction.
Therefore we only have two solutions: $x=2$ and $x=3$.
