Multiplying

Multiplying rational expressions works like $\frac{2}{3}\cdot \frac{9}{4}$: multiply the tops, multiply the bottoms, then cancel. Take $\frac{{x}^{2}-4}{3x}\cdot \frac{6x}{x-2}$. Factor first, so the shared factors are in plain view: ${x}^{2}-4$ is $\left(x+2\right)\left(x-2\right)$. Then multiply across:

$$\frac{\left(x+2\right)\left(x-2\right)\cdot 6x}{3x\left(x-2\right)}$$

Now cancel every factor the top and bottom share: x - 2 goes, x goes, and 6 over 3 leaves 2. What remains is $2\left(x+2\right)$, and multiplying out the top gives the finished answer, $2x+4$. There is no common denominator to find: that is only needed when you add or subtract.

The excluded values still come from the original bottoms: here 0 and 2, even though both cancelled.