Sign problem on complex fractions

Ok. Here is my problem.

$\displaystyle \frac{\frac{1}{a-1}+1}{\frac{1}{a+1}-1}$

And my aproach

$\displaystyle =\frac{\frac{1}{a-1}+\frac{a-1}{a-1}}{\frac{1}{a+1}-\frac{a+1}{a+1}}$

$\displaystyle =\frac{\frac{1+a-1}{a-1}}{\frac{1-a+1}{a+1}}$

$\displaystyle =\frac{\frac{a}{a-1}}{\frac{-a}{a+1}}$

$\displaystyle =\frac{a}{a-1}*\frac{a+1}{-a}$

$\displaystyle =\frac{(a)(a+1)}{(-a)(a-1)}$

I'm having some trouble to solve for these signs...mmm...The book I'm using has the answers to odd problems only. This is an even one. But anyway, I want to have it right. The istruction says "Simplify as much as possible" And I've got the feeling this can be worked further. Any idea?

PS: I thought about multiplicating by -1, but then It would be the same as the numerator'd turn into negatives.