#### Answer

$x=\pm\sqrt{2}$

#### Work Step by Step

$1+\displaystyle \frac{1}{(x+1)(x+2)}=\frac{2}{x+1}+\frac{1}{x+2}$
We multiply through by $(x+1)(x+2)$:
$(x+1)(x+2)+1=2(x+2)+(x+1)$
And distribute:
$x^{2}+3x+2+1=2x+4+x+1$
And simplify:
$x^{2}+3x+2+1-2x-4-x-1=0$
$x^{2}-2=0$
$x=\pm\sqrt{2}$