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.

You cannot divide both sides by $(x-2)(x-3)$ if it equals $0$. When you divide both sides by it, you are ignoring the solutions $x=2,3$
$$(x-1)(x-2)(x-3)=(x-2)(x-3)(x-4)$$
A proper way to solve this is as follows
$$(x-1)(x-2)(x-3)-(x-4)(x-2)(x-3)=0$$$$[(x-1)-(x-4)](x-2)(x-3)=0$$
$$3(x-2)(x-3)=0$$
