For all positive \(x\), which of the following is equal to \(\displaystyle 2+ \frac{3x}{x+2} - \frac{6}{2x+4} \) ?

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$$ 2+ \frac{3x}{x+2} - \frac{6}{2x+4} = 2+ \frac{3x}{x+2} - \frac{6}{2(x+2)}$$ $$ = \frac{2(2)(x+2)}{2(x+2)} + \frac{3x(2)}{2(x+2)} - \frac{6}{2(x+2)} $$ $$ = \frac{4x+8}{2(x+2)} + \frac{6x}{2(x+2)} -\frac{6}{2(x+2)} $$ $$ = \frac{10x+2}{2(x+2)} $$ $$ = \frac{2(5x+1)}{2(x+2)} $$ $$ = \boxed{\frac{5x+1}{x+2}} $$