1. a question about an explanation in permutations

Hi guyz, this was an explanation that Soroban so kindly explained to me but i just have a question about it and would really appreciate it if someone could help me

This is the question and explanation

(1) The ratio of the number of arrangements of different objects taken at a time
to the number of arrangments of different objects taken at a time is .
Find the value of .
objects taken at a time: .

objects taken at a time: .

The ratio is: .

And we have: .

. . which simplifies to: .

Therefore: .

i am just very, very confused about how we get from this step :

which i do understand but i dont know how to get to this part

where does this come from? i dont understand how we got rid of the factorials and made the above equation with 2n+1 and n+1
if anyone could explain to me the steps between these steps i would REALLI appreciate it
i would also like to thank Soroban again for the wonderful explanation and that this is just my lack of understanding that is creating problems
thank you to anyone who can help me !

2. $\frac{(2n+2)!}{(2n)!} = \frac{(2n+2)(2n+1)(2n)(2n-1)...3.2.1}{(2n)(2n-1)(2n-2)...3.2.1}=(2n+2)(2n+1)$