Multiplication of Taylor series

The question is:

Use multiplication of Taylor series to find the quartic Taylor polynomial about 0 for the function:

$\displaystyle f(x)=\frac{sinx}{\sqrt{1+x}}$ evaluating the coefficients.

For sin x the standard Taylor series about 0 is:

$\displaystyle x-\frac{1}{3!}x^3+...$

For $\displaystyle \sqrt{1+x}$ this can be rearranged to fit the standard Talor series

$\displaystyle (1+x)^\alpha$, which is

$\displaystyle 1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+\frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3$ ...

Multiplying these two together the answer I get is

$\displaystyle \frac{1}{2}x+\frac{3}{8}x^2+\frac{5}{96}x^3+\frac{ 35}{128}x^4$

Does this look along the right lines? (Worried)