Made new thread because I got new expression to solve. I have tried to get answer by myself, but ended with different result.

Expression:

$\displaystyle (\frac{2x}{x+2}+\frac{2x}{6-3x}+\frac{8x}{x^2-4})\div\frac{4x^2+16x}{x-2}$

Answer:

$\displaystyle \frac{1}{3(x+4)}$

I have done this:

$\displaystyle (\frac{2x*-3(x-2)+2x(x+2)+8x*-3}{-3(x+2)(x-2)})$

$\displaystyle (\frac{2x*-3(x-2)+2x(x+2)+8x*-3 *(x-2)}{-3(x+2)(x-2)*(4x^2+16x)})$

Here (x-2) should cancel each other out which leaves:

$\displaystyle (\frac{2x*-3(x-2)+2x(x+2)+8x*-3}{-3(x+2)(4x^2+16x)})=\frac{24x^2-44x-4x^3+48}{48x^2-192x-12x^4+48x^3}$

Pretty nice, when I made x=1 and solved my gigantic simplification it gave same result as text book simplification: 1/15

My creation blows, where I have done mistakes? How this needs to be done?