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To solve for \( x \) in the given equation:

\[
\frac{2}{x^2 - 4} - \frac{1}{x + 2} = \frac{3}{x - 2}
\]

### Step 1: Factor the Denominator
Notice that \( x^2 - 4 \) is a difference of squares, which can be factored as:
\[
x^2 - 4 = (x + 2)(x - 2)
\]

So the equation becomes:
\[
\frac{2}{(x + 2)(x - 2)} - \frac{1}{x + 2} = \frac{3}{x - 2}
\]

### Step 2: Find a Common Denominator
The common denominator for all terms is \( (x + 2)(x - 2) \).

### Step 3: Rewrite Each Fraction with the Common Denominator
Rewrite each term using the common denominator:

1. The first term already has \( (x + 2)(x - 2) \) as its denominator:
   \[
   \frac{2}{(x + 2)(x - 2)}
   \]

2. The second term needs \( x - 2 \) in the denominator:
   \[
   \frac{1 \cdot (x - 2)}{(x + 2)(x - 2)} = \frac{x - 2}{(x + 2)(x - 2)}
   \]

3. The third term needs \( x + 2 \) in the denominator:
   \[
   \frac{3 \cdot (x + 2)}{(x + 2)(x - 2)} = \frac{3(x + 2)}{(x + 2)(x - 2)}
   \]

Now the equation becomes:
\[
\frac{2 - (x - 2)}{(x + 2)(x - 2)} = \frac{3(x + 2)}{(x + 2)(x - 2)}
\]

### Step 4: Simplify the Numerator
Simplify \( 2 - (x - 2) \):
\[
2 - x + 2 = 4 - x
\]

So the equation now is:
\[
\frac{4 - x}{(x + 2)(x - 2)} = \frac{3(x + 2)}{(x + 2)(x - 2)}
\]

### Step 5: Set the Numerators Equal
Since the denominators are the same, we can set the numerators equal to each other:
\[
4 - x = 3(x + 2)
\]

### Step 6: Expand and Solve for \( x \)
Expanding the right side:
\[
4 - x = 3x + 6
\]

Bring all terms involving \( x \) to one side:
\[
4 - 6 = 3x + x
\]
\[
-2 = 4x
\]
\[
x = -\frac{1}{2}
\]

### Answer
The correct answer is:
\[
x = -\frac{1}{2}
\]

Let me know if you would like further details or have any questions!

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Here are 5 related questions to deepen your understanding:

1. How do you solve rational equations with more complex denominators?
2. What are the conditions for factoring a difference of squares?
3. Why is it necessary to use a common denominator in equations like this?
4. How can you verify your answer in a rational equation?
5. What are some potential pitfalls to watch out for in rational equations?

**Tip:** Always check if the values you find for \( x \) make any denominators zero, as these would be excluded from the solution.

Solve for x: \(\frac{2}{x^2 - 4} - \frac{1}{x + 2} = \frac{3}{x - 2}\)

Learn how to solve the rational equation \(\frac{2}{x^2 - 4} - \frac{1}{x + 2} = \frac{3}{x - 2}\) by factoring the difference of squares and finding a common denominator. This example demonstrates algebraic techniques for solving equations with rational expressions, suitable for high school students in Grades 9-12.

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