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The given equation is:

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

### Step-by-step solution:

1. **Factor the denominator \( x^2 - 1 \)**:
   \[
   x^2 - 1 = (x - 1)(x + 1)
   \]
   So the equation becomes:
   \[
   \frac{5}{(x - 1)(x + 1)} + \frac{2}{x + 1} = 2
   \]

2. **Combine the fractions on the left-hand side**:
   To combine the fractions, the common denominator is \( (x - 1)(x + 1) \).

   The second fraction needs to be rewritten to have the same denominator. Multiply the numerator and denominator of \( \frac{2}{x+1} \) by \( (x - 1) \):
   \[
   \frac{2}{x + 1} = \frac{2(x - 1)}{(x - 1)(x + 1)}
   \]

   Now, combine the fractions:
   \[
   \frac{5}{(x - 1)(x + 1)} + \frac{2(x - 1)}{(x - 1)(x + 1)} = \frac{5 + 2(x - 1)}{(x - 1)(x + 1)}
   \]

3. **Simplify the numerator**:
   Expand the numerator:
   \[
   5 + 2(x - 1) = 5 + 2x - 2 = 2x + 3
   \]

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

4. **Solve for \( x \)**:
   Multiply both sides of the equation by \( (x - 1)(x + 1) \) to eliminate the denominator:
   \[
   2x + 3 = 2(x - 1)(x + 1)
   \]
   Expand the right-hand side:
   \[
   2(x - 1)(x + 1) = 2(x^2 - 1) = 2x^2 - 2
   \]

   Now, the equation becomes:
   \[
   2x + 3 = 2x^2 - 2
   \]

   Rearrange the equation:
   \[
   2x^2 - 2 - 2x - 3 = 0
   \]
   Simplify:
   \[
   2x^2 - 2x - 5 = 0
   \]

5. **Solve the quadratic equation**:
   The quadratic equation is:
   \[
   2x^2 - 2x - 5 = 0
   \]
   Use the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \( a = 2 \), \( b = -2 \), and \( c = -5 \). Substituting these values:
   \[
   x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(2)(-5)}}{2(2)}
   \]
   \[
   x = \frac{2 \pm \sqrt{4 + 40}}{4}
   \]
   \[
   x = \frac{2 \pm \sqrt{44}}{4}
   \]
   \[
   x = \frac{2 \pm 2\sqrt{11}}{4}
   \]
   \[
   x = \frac{1 \pm \sqrt{11}}{2}
   \]

### Final solutions:
The two possible solutions for \( x \) are:
\[
x = \frac{1 + \sqrt{11}}{2} \quad \text{or} \quad x = \frac{1 - \sqrt{11}}{2}
\]

Would you like more details or have any questions?

### Related questions:
1. How do you verify the solutions of a quadratic equation?
2. What are the properties of the factored form of a quadratic expression?
3. How can the quadratic formula be derived from completing the square?
4. What is the significance of the discriminant in the quadratic formula?
5. How do you simplify radicals like \( \sqrt{44} \)?

### Tip:
Always check for extraneous solutions when solving rational equations by substituting the solutions back into the original equation.

Solve the equation: 5/(x^2 - 1) + 2/(x + 1) = 2.

Learn how to solve the rational equation 5/(x^2 - 1) + 2/(x + 1) = 2 with a detailed step-by-step solution. This problem covers factoring, combining fractions, and applying the quadratic formula. Suitable for students in Grades 10-12.

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