Mixed Numbers gone bad...

I've worked and re-worked this problem over the last couple of days and cannot understand where I'm going wrong. This is also the first time I'm attempting to use LaTeX so forgive me if I do it wrong.

Change the following in to common fractions:

$\displaystyle a^2+ab-b^2-\frac{a^3-2b^3}{a-2b}$

Right away I determine the LCD = a-2b

So I then re-write the equation:

$\displaystyle \frac{a^2(a-2b)}{a-2b}+\frac{ab(a-2b)}{a-2b}-\frac{b^2(a-2b)}{a-2b}-\frac{a^3-2b^3}{a-2b}$

After doing the multiplication I get:

$\displaystyle \frac{a^3-2a^2b+a^2b-2ab^2-ab^2+2b^3-a^3-2b^3}{a-2b}$

Combining like terms I get (I hope I've managed to keep all the information correct in the latex formatting):

$\displaystyle \frac{-a^2b-3ab^2}{a-2b}$

The answer the book shows is:

$\displaystyle -\frac{a^2b+3ab^2-4b^3}{a-2b}$

Any help would be greatly appreciated and much thanked in advance as always. Thanks!

E