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The equation shown in the image is:

\[
\frac{1}{x^2 - x - 6} + \frac{3}{x + 2} = \frac{-4}{x - 3}
\]

### Step-by-step solution:

1. **Factor the quadratic expression** \(x^2 - x - 6\):
   \[
   x^2 - x - 6 = (x - 3)(x + 2)
   \]
   So the equation becomes:
   \[
   \frac{1}{(x - 3)(x + 2)} + \frac{3}{x + 2} = \frac{-4}{x - 3}
   \]

2. **Find a common denominator** for the terms on the left-hand side:
   - The first term already has the denominator \((x - 3)(x + 2)\).
   - The second term can be rewritten to have the same denominator. Multiply the numerator and denominator by \((x - 3)\):
     \[
     \frac{3}{x + 2} = \frac{3(x - 3)}{(x - 3)(x + 2)} = \frac{3x - 9}{(x - 3)(x + 2)}
     \]

   Now the equation becomes:
   \[
   \frac{1 + (3x - 9)}{(x - 3)(x + 2)} = \frac{-4}{x - 3}
   \]
   Simplify the numerator on the left-hand side:
   \[
   \frac{3x - 8}{(x - 3)(x + 2)} = \frac{-4}{x - 3}
   \]

3. **Eliminate the denominator** by multiplying both sides by \((x - 3)\), assuming \(x \neq 3\):
   \[
   \frac{3x - 8}{x + 2} = -4
   \]

4. **Solve the resulting equation**:
   Multiply both sides by \(x + 2\) to eliminate the denominator:
   \[
   3x - 8 = -4(x + 2)
   \]
   Expand the right-hand side:
   \[
   3x - 8 = -4x - 8
   \]

5. **Simplify and solve for \(x\)**:
   Add \(4x\) to both sides:
   \[
   7x - 8 = -8
   \]
   Add 8 to both sides:
   \[
   7x = 0
   \]
   Divide by 7:
   \[
   x = 0
   \]

### Final solution:
The solution is \(x = 0\), with the restriction that \(x \neq 3\) and \(x \neq -2\) because these values would make the original denominators zero.

Would you like to see any more details or further clarifications?

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Here are 5 related questions:

1. What happens if we substitute \(x = 3\) or \(x = -2\) in the original equation?
2. Can you explain why we need a common denominator to solve the equation?
3. How do you factor quadratic expressions like \(x^2 - x - 6\)?
4. Why do we need to check for restrictions on \(x\) when solving rational equations?
5. How do you multiply and simplify rational expressions in algebra?

**Tip**: Always check for restrictions on variables in rational equations to avoid dividing by zero, which leads to undefined expressions.

Solve the rational equation: 1/(x^2 - x - 6) + 3/(x + 2) = -4/(x - 3)

Learn how to solve a rational equation involving factoring, finding a common denominator, and simplifying step-by-step. This problem covers rational equations, quadratic factoring, and algebraic manipulations suitable for students in Grades 9-12.

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