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

Summary Submit Skip Question

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