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The given expression is:

\[
\frac{x - 3}{x + 2} + \frac{4x}{x - 3} - \frac{8x + 4x^2}{x^2 - x - 6}
\]

### Step 1: Factor the quadratic in the third denominator

The denominator in the third term is \(x^2 - x - 6\). Let's factor it:

\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

So, the third term becomes:

\[
\frac{8x + 4x^2}{(x - 3)(x + 2)}
\]

### Step 2: Rewrite the full expression

Now, the full expression is:

\[
\frac{x - 3}{x + 2} + \frac{4x}{x - 3} - \frac{4x(2x + 1)}{(x - 3)(x + 2)}
\]

### Step 3: Find the common denominator

The least common denominator (LCD) is \((x + 2)(x - 3)\), so we need to express each term with this common denominator.

- First term: \(\frac{x - 3}{x + 2}\) becomes \(\frac{(x - 3)(x - 3)}{(x + 2)(x - 3)} = \frac{(x - 3)^2}{(x + 2)(x - 3)}\).
- Second term: \(\frac{4x}{x - 3}\) becomes \(\frac{4x(x + 2)}{(x + 2)(x - 3)}\).
- Third term: Already has the denominator \((x + 2)(x - 3)\), so it remains \(\frac{4x(2x + 1)}{(x + 2)(x - 3)}\).

### Step 4: Combine the numerators

Now, we have:

\[
\frac{(x - 3)^2 + 4x(x + 2) - 4x(2x + 1)}{(x + 2)(x - 3)}
\]

### Step 5: Simplify the numerator

1. Expand \((x - 3)^2 = x^2 - 6x + 9\).
2. Expand \(4x(x + 2) = 4x^2 + 8x\).
3. Expand \(4x(2x + 1) = 8x^2 + 4x\).

Now, substitute these back into the expression:

\[
(x^2 - 6x + 9) + (4x^2 + 8x) - (8x^2 + 4x)
\]

Simplifying:

\[
x^2 - 6x + 9 + 4x^2 + 8x - 8x^2 - 4x = -3x^2 - 2x + 9
\]

### Step 6: Final expression

The final expression is:

\[
\frac{-3x^2 - 2x + 9}{(x + 2)(x - 3)}
\]

Let me know if you'd like a more detailed breakdown of any part, or if you have any questions!

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Here are 5 questions that might help further explore the problem:

1. How do you factor quadratic expressions like \(x^2 - x - 6\)?
2. Why do we need to find a common denominator when adding or subtracting rational expressions?
3. Can the final expression be simplified further, or should it stay as is?
4. What happens to the expression if \(x = 3\) or \(x = -2\)?
5. How do you expand binomials like \((x - 3)^2\)?

**Tip:** Always check for values that make the denominator zero, as these are restrictions on the domain of the expression.

Simplify the expression: (x-3)/(x+2) + 4x/(x-3) - (8x+4x^2)/(x^2 - x - 6)

Learn how to solve rational expressions that involve factoring quadratic equations and finding a common denominator. This step-by-step guide simplifies a complex algebraic expression, using factoring, common denominators, and algebraic simplification techniques. Suitable for high school students.

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