If you have an equation a = b, and multiply by (x^2 - 1), the result will have a solution when x^2 - 1 = 0, that is if x = +/- 1, because your new equation is just 0=0 in that case.
You solve the new equation. Then you check manually if x = +/- 1 are actually solutions of the original equation. And if these two are not solutions of the new equation, that is highly suspicious and you need to check what is going on.
PS You multiplied by (x-1)^2, which is 0 if x=1 and in no other case. So you expect a solution x=1 to be added and no other.
PPS The rule is that you won’t lose solutions unless you divide both sides by zero. But you gain solutions if you multiply by zero.