$$\frac{ 1 }{ x+1 } + \frac{ 1 }{ x+2 } = \frac{ 2 }{ x+10 }$$
$x = -\frac{26}{17} = -1\frac{9}{17} \approx -1.529411765$
$$\left(x+2\right)\left(x+10\right)+\left(x+1\right)\left(x+10\right)=\left(x+1\right)\left(x+2\right)\times 2$$
$$x^{2}+12x+20+\left(x+1\right)\left(x+10\right)=\left(x+1\right)\left(x+2\right)\times 2$$
$$x^{2}+12x+20+x^{2}+11x+10=\left(x+1\right)\left(x+2\right)\times 2$$
$$2x^{2}+12x+20+11x+10=\left(x+1\right)\left(x+2\right)\times 2$$
$$2x^{2}+23x+20+10=\left(x+1\right)\left(x+2\right)\times 2$$
$$2x^{2}+23x+30=\left(x+1\right)\left(x+2\right)\times 2$$
$$2x^{2}+23x+30=\left(x^{2}+3x+2\right)\times 2$$
$$2x^{2}+23x+30=2x^{2}+6x+4$$
$$2x^{2}+23x+30-2x^{2}=6x+4$$
$$23x+30=6x+4$$
$$23x+30-6x=4$$
$$17x+30=4$$
$$17x=4-30$$
$$17x=-26$$
$$x=\frac{-26}{17}$$
$$x=-\frac{26}{17}$$
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