After factoring out the greatest common factor from $3x^2 - 12$, you will see $x^2 - 4$. Remember the formula for the difference of squares: $a^2 - b^2 = (a - b)(a + b)$.

Once the numerator is fully factored, see if any factor matches the denominator $x - 2$. If so, you can cancel that factor (as long as $x \ne 2$), then rewrite the remaining expression in simplest expanded form.

In Desmos, enter the expression $(3x^2 - 12)/(x - 2)$ on one line to see its graph. Notice the shape of the graph and that there is a hole (undefined point) at $x = 2$.

On separate lines, enter $y = 3x - 6$, $y = x^2 - 6$, $y = 3x + 4$, and $y = 3x + 6$. Each will appear as a different curve or line.

Turn each of the answer-choice graphs on and off, and see which one lies exactly on top of the graph of $(3x^2 - 12)/(x - 2)$ everywhere it is defined (except at the hole at $x = 2$). The expression whose graph overlaps the rational graph in this way is the correct choice.

Start by factoring the numerator $3x^2 - 12$.

Both terms have a factor of $3$:

So the expression becomes

Notice that $x^2 - 4$ is a difference of squares: $x^2 - 2^2$.

Use the identity $a^2 - b^2 = (a - b)(a + b)$:

Substitute this into the numerator:

Now there is a common factor $(x - 2)$ in the numerator and denominator.

For $x \ne 2$, you can cancel this common factor:

The original expression is undefined at $x = 2$, and the simplified expression $3(x + 2)$ represents the same values for all other $x$. The question only asks about values for which the expression is defined, so this cancellation is valid.

Expand $3(x + 2)$ by distributing $3$:

So an equivalent expression to $\dfrac{3x^2 - 12}{x - 2}$, for all $x$ where it is defined, is $3x + 6$, which corresponds to choice D.