STRATEGY: As with all GRE Multiple Choice questions, we should immediately ask ourselves, Can I use the answer choices to my advantage?
In this case, we can easily test the answer choices.
Now we should give ourselves about 20 seconds to identify a faster approach.
In this case, we can also solve the equation.
Since there aren't many values (among the answer choices) to test, I'm going to test the given values.

Test $x = -1$ by plugging it into the original equation to get: $\frac{(-1)-2}{(-1)^2-4} =1$
Simplify: $\frac{-3}{-3} =1$.
WORKS!
This means we can eliminate answer choice B, since it doesn't include $x = -1$ as a solution.

Now test $x = 2$ by plugging it into the original equation to get: $\frac{2-2}{2^2-4} =1$
Simplify: $\frac{0}{0} =1$.
Doesn't work.
This means we can eliminate answer choices A and E, since they stated that $x = 2$ is a solution.

Test $x = -3$ by plugging it into the original equation to get: $\frac{(-3)-2}{(-3)^2-4} =1$
Simplify: $\frac{-5}{5} =1$.
Doesn't work.
This means we can eliminate answer choice C, since it states that $x = -3$ is a solution.

By the process of elimination, the correct answer is D