Factoring Algebra Expression

Question

I have the below algebra expression:

$$ (x-1)((x-1)^2 - 1) = 6y$$

I'm trying to get the left hand side be $x(x^2-1)$: $$ x(x^2-1) = 6y \dots$$ Attempt: $$ (x-1)(x^2-2x) = 6y$$ $$ x^3 - 3x^2 + 2x = 6y$$ I'm stuck here

Answer 1

Do not worry, your algebra is correct.

The problem is that $$(x-1)((x-1)^2 - 1) = (x-1)(x^2 -2x)=x(x-1)(x-2)$$

is not the same as $$x(x^2-1)=x(x-1)(x+1)$$

Answer 2

but it is $$x(x-1)(x+1)=(x-1)(x+1)x=(x-1)(x^2+x)\ne (x-1)(x^2-2x+1-1)$$