Factorial equation: why does $n!\cdot(n + 2)! = n\cdot(n + 2)!\cdot(n - 1)!$

Question

Can someone explain how this:

$$n!\cdot(n + 2)!$$

could become:

$ n\cdot(n + 2)!\cdot(n - 1)!$

Answer 1

$$ n! = n \cdot (n - 1)! $$ That's all.

Answer 2

You have

$$n!=1\times 2\times\cdots\times n$$

so

$$n!=(1\times\cdots \times n-1)\times n=(n-1)!\times n.$$

Therefore

$$n!\cdot (n+2)!=n\cdot (n-1)!\cdot (n+2)! = n\cdot (n+2)!\cdot (n-1)!.$$

And that's it.