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The given equation is:

\[
\frac{1}{x - 4} - \frac{5}{x + 2} = \frac{6}{x^2 - 2x - 8}
\]

### Step 1: Factor the quadratic denominator on the right side

We need to factor \( x^2 - 2x - 8 \):

\[
x^2 - 2x - 8 = (x - 4)(x + 2)
\]

So, the equation becomes:

\[
\frac{1}{x - 4} - \frac{5}{x + 2} = \frac{6}{(x - 4)(x + 2)}
\]

### Step 2: Combine the fractions on the left-hand side

The least common denominator (LCD) for the fractions on the left-hand side is \( (x - 4)(x + 2) \). Let's rewrite both fractions:

\[
\frac{1}{x - 4} = \frac{(x + 2)}{(x - 4)(x + 2)}
\]
\[
\frac{5}{x + 2} = \frac{5(x - 4)}{(x - 4)(x + 2)}
\]

Now, substitute these into the equation:

\[
\frac{x + 2}{(x - 4)(x + 2)} - \frac{5(x - 4)}{(x - 4)(x + 2)} = \frac{6}{(x - 4)(x + 2)}
\]

### Step 3: Simplify the left-hand side

Now, simplify the left-hand side:

\[
\frac{x + 2 - 5(x - 4)}{(x - 4)(x + 2)} = \frac{6}{(x - 4)(x + 2)}
\]

Simplify the numerator:

\[
x + 2 - 5(x - 4) = x + 2 - 5x + 20 = -4x + 22
\]

So the equation becomes:

\[
\frac{-4x + 22}{(x - 4)(x + 2)} = \frac{6}{(x - 4)(x + 2)}
\]

### Step 4: Cancel the denominators

Since the denominators are the same, we can cancel them out:

\[
-4x + 22 = 6
\]

### Step 5: Solve for \(x\)

Now solve for \(x\):

\[
-4x + 22 = 6
\]
\[
-4x = 6 - 22
\]
\[
-4x = -16
\]
\[
x = 4
\]

### Step 6: Check for extraneous solutions

We must check if \( x = 4 \) is a valid solution by substituting it back into the original equation. However, if you substitute \( x = 4 \), you will notice that it makes the denominators \( x - 4 \) zero, which is undefined.

Therefore, **there is no solution** to this equation.

Would you like more details or have any questions?

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Here are 5 related questions for practice:

1. How would you solve similar rational equations with higher-degree polynomials?
2. What is the general process to find the least common denominator (LCD) of rational expressions?
3. How do you factor quadratic expressions quickly and accurately?
4. What are extraneous solutions, and why do they occur in rational equations?
5. How do you determine the domain of a rational function?

**Tip:** Always check the domain of the variable when solving rational equations to avoid extraneous solutions.

Solve the equation: (1 / (x - 4)) - (5 / (x + 2)) = 6 / (x^2 - 2x - 8).

This problem involves solving a rational equation by factoring the quadratic expression and combining fractions with a common denominator. Learn the step-by-step process to solve (1 / (x - 4)) - (5 / (x + 2)) = 6 / (x^2 - 2x - 8) and check for extraneous solutions.

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