$-\dfrac{1}{a+b} + -\dfrac{1}{a-b}$

$\left(a^{2}-b^{2}\right) x^{2}+2 a x+1=0$

$x_{1,2}=\dfrac{-2 a \pm \sqrt{(2 a)^{2}-4\left(a^{2}-b^{2}\right)}}{2\left(a^{2}-b^{2}\right)}$

$x_{1,2}=\dfrac{-2 a \pm \sqrt{4 a^{2}-4 a^{2}+4 b^{2}}}{2\left(a^{2}-b^{2}\right)}$

$x_{1,2}=\dfrac{-2 a \pm \sqrt{4 b^{2}}}{2\left(a^{2}-b^{2}\right)}$

$x_{1,2}=\dfrac{-2 a \pm 2 b}{2\left(a^{2}-b^{2}\right)}$

$x_{1,2}=\dfrac{-a \pm b}{a^{2}-b^{2}}=\dfrac{-a \pm b}{(a-b)(a+b)}$

$x_{1}=\dfrac{-a+b}{(a-b)(a+b)}=\dfrac{-(a-b)}{(a-b)(a+b)}=-\dfrac{1}{(a+b)}$

$x_{2}=\dfrac{-a-b}{(a-b)(a+b)}=\dfrac{-(a+b)}{(a-b)(a+b)}=-\dfrac{1}{(a-b)}$