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To solve the given equation:

\[
\frac{1}{x+3} + \frac{5}{x^2 - 9} = \frac{2}{x - 3}
\]

we will follow these steps:

1. **Factorize the Denominator \(x^2 - 9\)**:
   \[
   x^2 - 9 = (x + 3)(x - 3)
   \]

2. **Rewrite the Equation**:
   Substituting the factorized form, the equation becomes:
   \[
   \frac{1}{x+3} + \frac{5}{(x+3)(x-3)} = \frac{2}{x-3}
   \]

3. **Find a Common Denominator**:
   The common denominator for all terms is \((x + 3)(x - 3)\). Rewriting each term with this denominator:
   \[
   \frac{(x - 3)}{(x + 3)(x - 3)} + \frac{5}{(x + 3)(x - 3)} = \frac{2(x + 3)}{(x + 3)(x - 3)}
   \]

   This simplifies to:
   \[
   \frac{x - 3 + 5}{(x + 3)(x - 3)} = \frac{2(x + 3)}{(x + 3)(x - 3)}
   \]

   Further simplifying the numerator on the left side:
   \[
   \frac{x + 2}{(x + 3)(x - 3)} = \frac{2(x + 3)}{(x + 3)(x - 3)}
   \]

4. **Equate the Numerators**:
   Since the denominators are equal, we equate the numerators:
   \[
   x + 2 = 2(x + 3)
   \]

   Expanding the right side:
   \[
   x + 2 = 2x + 6
   \]

5. **Solve for \(x\)**:
   Rearrange the equation:
   \[
   2 - 6 = 2x - x
   \]
   Simplifying:
   \[
   -4 = x \quad \text{or} \quad x = -4
   \]

6. **Check for Restrictions**:
   The restrictions on \(x\) are that \(x \neq \pm 3\), because they would make the denominators zero.

Hence, the solution is:
\[
x = -4
\]

Would you like any further details or explanations on this solution? Here are five related questions:

1. How do you find the least common denominator for multiple rational expressions?
2. What are the steps to factor quadratic expressions like \(x^2 - 9\)?
3. Can this method be applied to solve equations with more complex rational expressions?
4. How do you handle cases where the solution makes the denominator zero?
5. What are the implications of extraneous solutions in rational equations?

**Tip:** Always verify your solutions by substituting them back into the original equation to check for any contradictions or undefined values.

Explore the step-by-step solution to rational expressions involving factoring and simplifying with the equation 1/(x+3) + 5/(x^2-9) = 2/(x-3). This solution demonstrates how to factorize quadratics and simplify expressions to find the value of x, suitable for high school students studying algebra.

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