<s>[INST] Show that $\frac {1}{\sqrt{5}}[(\frac {1}{x+r_+}) - (\frac {1}{x+r_-}) = \frac {1}{\sqrt{5}x}[(\frac {1}{1-r_{+}x}) - (\frac {1}{1-r_{-}x})] $ I need to manipulate this equation:
$$
\frac {1}{\sqrt{5}}\left(\frac {1}{x+r_+} - \frac {1}{x+r_-}\right) 
$$
to show that
$$ \frac {1}{\sqrt{5}}\left(\frac {1}{x+r_+} - \frac {1}{x+r_-}\right) = \frac {1}{\sqrt{5}x}\left(\frac {1}{1-r_{+}x} -\frac {1}{1-r_{-}x}\right)
$$ (*)
where$$r_+=\frac {1+\sqrt{5}}{2} $$and $$r_-=\frac {1-\sqrt{5}}{2}$$ 
I know that both RHS and LHS of (*) are equal to $\frac{1}{1-x-x^2}  $ , but don't know how we get RHS manipulating LHS before someone told us that RHS is also equal to $\frac{1}{1-x-x^2} $ 
Edit:
Since $$r_+r_-=-1$$, I replace $r_+$ with $r_-$ and $r_-$ with $ r_+$ in the LHS of (*)
So that, I get
$$ \frac {1}{\sqrt{5}}\left(\frac {r_+}{1-r_+x} - \frac {r_-}{1-r_-x}\right)$$ Still, I am not there. Can anybody help me at this step?
After showing the equality above I can continue as
$$
\left(\frac {1}{1-r_{+}x}\right) - \left(\frac {1}{1-r_{-}x}\right)= {\sum_{n\ge0}\ r_+^nx^n}-{\sum_{n\ge0}\ r_-^nx^n}
$$
 [/INST] \begin{align*}
\frac {1}{\sqrt{5}}\left(\frac {1}{x+r_+} - \frac {1}{x+r_-}\right) 
    &= \frac {1}{\sqrt{5}} \frac{x}{x} \left(\frac {1}{x+r_+} - \frac  {1}{x+r_-}\right)  &&,x \neq 0  \\
    &= \frac {1}{\sqrt{5}\, x} \left(\frac {x}{x+r_+} - \frac  {x}{x+r_-}\right)  &&,x \neq 0  \\
    &= \frac {1}{\sqrt{5}\, x} \left(\frac {(x^2 + r_- x) - (x^2 + r_+ x)}{(x+r_+)(x+r_-)}\right)  &&,x \neq 0  \\
    &= \frac {1}{\sqrt{5}\, x} \left(\frac {r_- x - r_+ x}{(x+r_+)(x+r_-)}\right)  &&,x \neq 0  \\
    &= \frac {1}{\sqrt{5}\, x} \left(\frac {r_- x - r_+ x}{x^2 + x - 1}\right)  &&,x \neq 0  \\
    &= \frac {1}{\sqrt{5}\, x} \left(\frac {r_- x - r_+ x}{-(-x^2 - x + 1)}\right)  &&,x \neq 0  \\
    &= \frac {1}{\sqrt{5}\, x} \left(\frac {-r_- x + r_+ x}{-x^2 - x + 1}\right)  &&,x \neq 0  \\
    &= \frac {1}{\sqrt{5}\, x} \left(\frac {1-r_- x -1 + r_+ x}{(1-r_+ x)(1-r_- x)}\right)  &&,x \neq 0  \\
    &= \frac {1}{\sqrt{5}\, x} \left(\frac {(1-r_- x) - (1 - r_+ x)}{(1-r_+ x)(1-r_- x)}\right)  &&,x \neq 0  \\
    &= \frac {1}{\sqrt{5}\, x} \left(\frac {1}{1-r_+ x} - \frac {1}{1-r_- x}\right)  &&,x \neq 0  \\
\end{align*}
For completeness, we verify that the RHS of the equation sought is undefined for $x=0$, so the condition cannot be removed.
</s>