$$\frac{ 2x+2 }{ x-1 } - \frac{ 6 }{ x } =2$$
$x=3$
$$x\left(2x+2\right)-\left(x-1\right)\times 6=2x\left(x-1\right)$$
$$2x^{2}+2x-\left(x-1\right)\times 6=2x\left(x-1\right)$$
$$2x^{2}+2x-\left(6x-6\right)=2x\left(x-1\right)$$
$$2x^{2}+2x-6x+6=2x\left(x-1\right)$$
$$2x^{2}-4x+6=2x\left(x-1\right)$$
$$2x^{2}-4x+6=2x^{2}-2x$$
$$2x^{2}-4x+6-2x^{2}=-2x$$
$$-4x+6=-2x$$
$$-4x+6+2x=0$$
$$-2x+6=0$$
$$-2x=-6$$
$$x=\frac{-6}{-2}$$
$$x=3$$
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