urgent* binomial series help(work included)

Use the binomial series to expand the function as a power series.

1 / (2+x)^3

the answer in the back says it's (-1)^n(n+1)(n+2) * x^n all over 2^n+4

can someone help me where i went wrong or if im doing it totally wrong

My work:

[2(1+x/2)]^-3 = (1/8)(1+x/2)^-3

(1/8)*sum (-3 n)(x/2)^n

(1/8) [ 1+(-3)(x/2) + [(-3)(-4)]/2 * (x/2)^2 +... (-3 + n - 1) ]

(1/8) * sum [(-1)^n (3*4*5....n(n+1) ]/ n!

(1/2^3) * [ (-1)^n(n+1)(n+2) ] / 2^n

=[[ (-1)^n * (n+1)(n+2) ]x^n ]/ (2^n+3) ]

thanks.