Shifting the index of $\sum_{n=2}^{\infty}a_n\cdot n\cdot (n-1)\cdot x^{n-2}$ I want to shift the index of 
$$\sum_{n=2}^{\infty}a_n\cdot n\cdot (n-1)\cdot x^{n-2}$$
to start at $n=0$. If I expand this series I get:

$$a_2\cdot 2\cdot 1\cdot x^0+a_3\cdot 3\cdot 2\cdot x^1+\cdots$$

But if I start simply at $n=0$ without doing any modification, I get:

$$a_0\cdot 0\cdot (-1)\cdot x^{-2}+a_1\cdot 1\cdot0\cdot x^{-1}+a_2\cdot2\cdot1\cdot x^0+\cdots$$

which is the same as the series above, because the first $2$ terms are $0$, so I tought I could just write it like this:

$$\sum_{n=0}^{\infty}a_n\cdot n\cdot (n-1)\cdot x^{n-2}$$

but My book writes it as the following:

$$\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n$$

what's happening?
Both are correct. In the book, they just "shifted" the whole expression. 

They just replaced $n$ with $n+2$.
So instead of starting with $n=2$ they start with $n=0$, but they use $n+2$ in the expression, so it is the same as just using $n=2$ in the original formula.

If you expand both series, you will see it is the same.
Nothing happens ! 

$\sum_{n=0}^{\infty}a_n\cdot n\cdot (n-1)\cdot x^{n-2}=\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n$