Final answer:
The provided options (A-D) don't match the simplified form of the given expression ((n!)(n+4)!)/((n+2)!(n-2)!) which simplifies to (n+4)(n+3). This suggests a typo in the question or options.
Step-by-step explanation:
The question asks us to simplify the expression ((n!)(n+4)!)/((n+2)!(n-2)!). To simplify a ratio of factorials like this, we need to factorize and cancel out common terms.
The factorial (n+4)! can be expanded as (n+4)(n+3)(n+2)(n+1)n!. The (n+2)! in the denominator expands as (n+2)(n+1)n!. When we cancel the common terms n! we are left with (n+4)(n+3)/(n+2). Further canceling (n+2), we get (n+4)(n+3)/(n+2) in apparent contraction with the provided options.
In the given options A through D, however, none of them match the correct expression we derived, which suggests there might be an error in the original expression or in the options provided. If the question intended to simplify ((n!)(n+4)!)/((n+2)!(n-2)!), then it would simplify to (n+4)(n+3), which is not listed among the options A-D. It's possible the original question or options may have a typo.