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I'm sure this is basic but I am having a mind blank.

I need to simply as much as possible and these are my steps so far:

$$\frac{(2n)!((n+1)!)^2a^{2n+2}}{(2n+2)!(n!)^2a^{2n}} = \frac{(2n)!((n+1)!)^2a^{2}}{(2n+2)!(n!)^2} = \frac{(n+1)!^2}{(2n+2)(2n+1)n!^2}a^{2}$$

I think this is right so far but not sure where to go on cancelling factorials and so on. Any insight would be great.

Thanks

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$$\frac{(n+1)!^2}{(2n+2)(2n+1)n!^2}a^{2} = \frac{n!(n+1)!}{2(2n+1)n!n!}a^2 =\frac{a^2(n+1)}{2(2n+1)}$$

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