Simplify the following factorial expression: (2n + 2)! / (2n)!

We are tasked with simplifying the factorial expression:

(2n + 2)! / (2n)!

Step-by-step solution:

Write out the factorials: (2n + 2)! means the product of all integers from 1 to (2n + 2). (2n)! means the product of all integers from 1 to (2n).

So, we can write: (2n + 2)! = (2n + 2) * (2n + 1) * (2n)!

Substitute into the original expression: Now, substitute this expression into the original factorial expression: (2n + 2)! / (2n)! = [(2n + 2) * (2n + 1) * (2n)!] / (2n)!

Cancel out the common (2n)! terms: The (2n)! terms in the numerator and denominator cancel out, leaving: (2n + 2) * (2n + 1)

Final simplified expression: Therefore, the simplified expression is: (2n + 2) * (2n + 1)

Answer: (2n + 2)(2n + 1)