Find the power series $f(x) = 4/(x+2)$

We know the geometric series:

$$\sum\_{n=1}^{\infty} x^{n-1} = \frac{1}{1-x}$$

$(x+2) = 1 - (-x - 1)$

So:

$$\sum\_{n=1}^{\infty} (-1)^{n-1}\cdot(x + 1)^{n-1} = \frac{1}{x + 2}$$

Multiply by $4$

$$\sum\_{n=1}^{\infty} 4(-1)^{n-1} \cdot (x+1)^{n+1} = \frac{4}{x+2}$$

But this isnt the correct answer as the book points out. why?