See picture.
$\displaystyle \dfrac{a+b}{c} = \dfrac{a}{c} + \dfrac{b}{c}$
This means
$\displaystyle \dfrac{(x+1)\dfrac{-1}{\sqrt{1-x^2}}-\cos^{-1}(x)}{(x+1)^2} = -\dfrac{(x+1)\dfrac{1}{\sqrt{1-x^2}}}{(x+1)^2} - \dfrac{\cos^{-1}(x)}{(x+1)^2}$
In the first fraction, multiply top and bottom by $\displaystyle \dfrac{\sqrt{1-x^2}}{x+1}$:
$\displaystyle -\dfrac{(x+1)\dfrac{1}{\sqrt{1-x^2}}}{(x+1)^2}\cdot \dfrac{\dfrac{\sqrt{1-x^2}}{x+1}}{\dfrac{\sqrt{1-x^2}}{x+1}} - \dfrac{\cos^{-1}(x)}{(x+1)^2} = -\dfrac{1}{(x+1)\sqrt{1-x^2}} - \dfrac{\cos^{-1}(x)}{(x+1)^2}$
There are lots of strategies for simplifying expressions. At this point, I do it by rote. When I first learned, though, I made a lot of careless mistakes. But, with practice, it became second nature. So, I would recommend finding an intermediate algebra book (any book used for an Algebra II class) and practicing the problems. If you do even one problem a day for a few weeks, I suspect you will see a marked improvement in your algebra skills.