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Let's solve the equation step by step.

The given equation is:

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

### Step 1: Eliminate fractions by multiplying the entire equation by \( x(x - 6) \).

Multiplying both sides by \( x(x - 6) \):

\[
x(x - 6)\left( \frac{8}{x} + 2 \right) = x(x - 6)\left( \frac{x + 4}{x - 6} \right)
\]

Simplifying:

\[
(x - 6) \cdot 8 + 2x(x - 6) = x(x + 4)
\]

\[
8(x - 6) + 2x(x - 6) = x(x + 4)
\]

Expanding both sides:

\[
8x - 48 + 2x^2 - 12x = x^2 + 4x
\]

### Step 2: Combine like terms.

On the left side:

\[
8x - 12x = -4x
\]
So the left side becomes:

\[
2x^2 - 4x - 48 = x^2 + 4x
\]

### Step 3: Move all terms to one side of the equation.

\[
2x^2 - 4x - 48 - x^2 - 4x = 0
\]

Simplifying:

\[
x^2 - 8x - 48 = 0
\]

### Step 4: Solve the quadratic equation.

Use the quadratic formula:

\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(-48)}}{2(1)}
\]

\[
x = \frac{8 \pm \sqrt{64 + 192}}{2}
\]

\[
x = \frac{8 \pm \sqrt{256}}{2}
\]

\[
x = \frac{8 \pm 16}{2}
\]

So:

\[
x = \frac{8 + 16}{2} = 12 \quad \text{or} \quad x = \frac{8 - 16}{2} = -4
\]

### Step 5: Exclude extraneous solutions.

From the note, \( x \neq 0 \) and \( x \neq 6 \). Neither 12 nor -4 violates this condition, so both solutions are valid.

### Final Answer:

The solution set is \(\{ -4, 12 \}\), so the correct answer is:

\[
\boxed{\{ -4, 12 \}}
\]

Would you like more details on any of the steps? Here are five related questions:

1. How do you factor quadratic equations?
2. Can you explain the quadratic formula in more detail?
3. How do you know when to exclude extraneous solutions in rational equations?
4. How do you handle equations with fractions like this one?
5. What other methods could solve quadratic equations?

**Tip:** Always check for restrictions when solving rational equations to avoid extraneous solutions.

What is the solution set of the equation 8/x + 2 = (x + 4)/(x - 6)? Note: x ≠ {0, 6}

Step-by-step solution for solving the rational equation 8/x + 2 = (x + 4)/(x - 6), using algebraic techniques and the quadratic formula. Includes tips on handling restrictions in rational equations. Suitable for Grades 9-10.

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