$$\frac{ 1 }{ x+1 } + \frac{ 1 }{ x+4 } = \frac{ 2 }{ x+3 }$$
$x=-7$
$$\left(x+3\right)\left(x+4\right)+\left(x+1\right)\left(x+3\right)=\left(x+1\right)\left(x+4\right)\times 2$$
$$x^{2}+7x+12+\left(x+1\right)\left(x+3\right)=\left(x+1\right)\left(x+4\right)\times 2$$
$$x^{2}+7x+12+x^{2}+4x+3=\left(x+1\right)\left(x+4\right)\times 2$$
$$2x^{2}+7x+12+4x+3=\left(x+1\right)\left(x+4\right)\times 2$$
$$2x^{2}+11x+12+3=\left(x+1\right)\left(x+4\right)\times 2$$
$$2x^{2}+11x+15=\left(x+1\right)\left(x+4\right)\times 2$$
$$2x^{2}+11x+15=\left(x^{2}+5x+4\right)\times 2$$
$$2x^{2}+11x+15=2x^{2}+10x+8$$
$$2x^{2}+11x+15-2x^{2}=10x+8$$
$$11x+15=10x+8$$
$$11x+15-10x=8$$
$$x+15=8$$
$$x=8-15$$
$$x=-7$$
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