TITLE: What are the rules for factorial manipulation?
QUESTION [9 upvotes]: I know that 
$$(k+1)! - 1 + (k+1)(k+1)! = (k+2)! - 1$$
thanks to wolframalpha, but I don't understand the steps for simplification, and I can't seem to find any rules about factorial manipulations on google.  Can someone explain this please?

REPLY [3 votes]: Assuming we are given only the task: 

Simplify the following: $\quad (k+1)! - 1 + (k+1)(k+1)!$

Let $\color{blue}{\bf x = k+1}$. Then we have 
$$
\begin{align}
x! - 1 + x(x!) 
& = x\;(x!) + 1\cdot x! -1 \\ \\
& = (x + 1)\;x! - 1 \quad\quad\quad\quad\quad\quad\tag{distributive property over addition} \\ \\
& = (\color{blue}{\bf x} + 1)! - 1 \\ \\
& =  (\color{blue}{\bf k+1} + 1)! - 1 \\ \\
& = (k+2)! - 1
\end{align}
$$