$$\frac{2x^{2}+4x+2}{x+1}\div\frac{2}{x}$$
$x\left(x+1\right)$
$$\frac{\left(2x^{2}+4x+2\right)x}{\left(x+1\right)\times 2}$$
$$\frac{2x\left(x+1\right)^{2}}{2\left(x+1\right)}$$
$$x\left(x+1\right)$$
$$x^{2}+x$$
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$x^{2}+x$
