Im not sure what ! is called but my question is how is

(x+1)! / x! = (x+1)

also is there a formula to solve

(x+1)^a where a equals a # (for example 18)

In general is x is a real number with x>-1 [so that it isn't neccessarly an integer...] the factorial function [or 'Pi function' according to Gauss interpretation...] is defined as a definite integral that I don't report because the details are a little complex. If You want more details about it see...

Gamma function - Wikipedia, the free encyclopedia
In particular the 'facrorial function' satisfies the relation...

\(\displaystyle x! = x\ (x-1)!\) (1)

Till now we are 'all right'... a little less clear for Me is the second part of Your question, regarding the expression \(\displaystyle (x+1)^{a}\)... could You supply more details please?...

Kind regards

\(\displaystyle \chi\) \(\displaystyle \sigma\)