Help in factorization (Explain if I was wrong)

Question

Task: Factorize $ax^2-a-x^2+x$.

And my answer is:

$$ a(x^2-1)-x(x-1)=a(x-1)(x+1)-x(x-1)$$

Am I right? If not, explain why and give the correct solution.

Answer 1

Unfortunatly, not correct. I could be downvoted

$$ax^2-a-x^2+x=$$ $$a(x^2-1)-(x^2-x)=$$ $$a(x+1)\color{red}{(x-1)}-x\color{red}{(x-1)}=$$

$$\color{red}{(x-1)}\Bigl(a(x+1)-x\Bigr)$$

Answer 2

"Factoring" usually means to write the whole expression as a product, not just factoring some terms.

Can see that $(x-1)$ can be factored out of your last answer now?