I have a set of convergence problems that I know I should solve with a ratio test,

but in denominator I have either $\displaystyle 2n!$ or $\displaystyle (2n+2)!$ or even $\displaystyle (n+1)!$ and I am little bit shakey with expanding these factorials for cancellation. I am only familiar with 5! or n! but not with factorial of sums or factorials of sums of products of variables and integers.

Is there a rule or a formula how could I do so? Would someone please show me how is and why $\displaystyle (2n+2)!$ expanded.

Thank you.