Proof by Induction simplification.

Question

I'm doing a proof by induction, this is the answer provided in the textbook. I reached the second to last step but I'm not sure how they simplified to the last step?

= (k+1)!−1+(k+1)·(k+1)!
= (k + 1)!(k + 2) − 1 = (k + 2)! − 1.

Answer 1

$$ (k+1)!−1+(k+1)·(k+1)! $$

Take the $-1$ to the end of the list and you have

$$= (k+1)!+(k+1)·(k+1)!-1$$

Now factor $(k+1)!$ to get $$ (k + 1)![1+(k +1)]− 1$$

$$= (k + 1)!(k + 2) − 1 = (k + 2)! − 1$$

Done!