Simplyfying factorials why is $(n+1)(n-1)!-(n-1)! = n(n-1)!$?

Question

I am a bit unclear on how these two expressions are equal:

$$(n+1)(n-1)!-(n-1)! = n(n-1)!$$

So far, I obtained

$$\frac{(n+1)!}{n}-(n-1)!=\frac{(n+1)n(n-1)!}{n}-(n-1)! = (n+1)(n-1)!-(n-1)!$$

However, I am unsure about the last step.

Answer 1

If you're not seeing it, set $N=(n-1)!$

Then we have:

$$N(n+1) - N = N(n+1-1)=\cdots$$

Answer 2

You are fine. Just complete the argument by noting that $(n+1)(n-1)!-(n-1)!=(n+1-1)(n-1)!=n(n-1)!$.