After you replace $x - \dfrac{1}{x}$ with a single symbol, what does the numerator look like? Can you recognize it as a standard factoring pattern, such as a difference of squares?

Once you factor the numerator, compare it to the denominator. Is there a common factor that you can cancel? After cancelling, what remains?

After simplifying with your temporary variable, remember to replace it by $x - \dfrac{1}{x}$ and see which answer choice matches your final simplified form.

In Desmos, type the original function, for example as f(x) = ((x-1/x)^2 - 4)/(x-1/x + 2). Notice that the graph may have holes or undefined points where $x=0$ or the denominator is zero.

Enter each option as a separate function, such as g(x) = x-1/x+2, h(x) = x+1/x-2, etc. Make sure they are all visible on the same coordinate plane.

Look for which choice’s graph lies exactly on top of the graph of the original function everywhere the original is defined (it is okay if the original has missing points where the simplified graph is continuous). The option whose graph perfectly matches in this way is the equivalent expression.

Notice that $x-\dfrac{1}{x}$ appears both inside the square in the numerator and (almost) as the whole denominator.

Let

Then the original expression becomes

The numerator $\nu^2 - 4$ is a difference of squares, since $4 = 2^2$.

Use the identity $a^2 - b^2 = (a-b)(a+b)$ with $a = \nu$ and $b = 2$:

So the whole expression is

As long as $\nu + 2 \neq 0$ (which is exactly when the original denominator is not zero), we can cancel the common factor $\nu + 2$ in the numerator and denominator:

The problem already restricts us to values of $x$ for which the original expression is defined, so this cancellation is valid on that domain.

Recall $\nu = x - \dfrac{1}{x}$, so

Thus the expression is equivalent to $x - \dfrac{1}{x} - 2$, which corresponds to answer choice D.