4.) (3x²+11x-4/x²+2x-8)(x-2/x+2)

5.) (x²-9x/x²+3x-10)(x²-4/x²-x-6)

Let's solve both expressions step by step.

### Expression 4:

\[
\left(\frac{3x^2 + 11x - 4}{x^2 + 2x - 8}\right)\left(\frac{x - 2}{x + 2}\right)
\]

**Step 1: Factor the quadratic expressions**

1. \( 3x^2 + 11x - 4 \):
   \[
   3x^2 + 11x - 4 = (3x - 1)(x + 4)
   \]

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

So, the expression becomes:

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

**Step 2: Cancel out common factors**

- The factor \( (x + 4) \) cancels out.
- The factor \( (x - 2) \) cancels out.

The simplified expression is:

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

### Expression 5:

\[
\left(\frac{x^2 - 9x}{x^2 + 3x - 10}\right)\left(\frac{x^2 - 4}{x^2 - x - 6}\right)
\]

**Step 1: Factor the quadratic expressions**

1. \( x^2 - 9x \):
   \[
   x^2 - 9x = x(x - 9)
   \]

2. \( x^2 + 3x - 10 \):
   \[
   x^2 + 3x - 10 = (x + 5)(x - 2)
   \]

3. \( x^2 - 4 \):
   \[
   x^2 - 4 = (x - 2)(x + 2) \quad \text{(Difference of squares)}
   \]

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

So, the expression becomes:

\[
\left(\frac{x(x - 9)}{(x + 5)(x - 2)}\right)\left(\frac{(x - 2)(x + 2)}{(x - 3)(x + 2)}\right)
\]

**Step 2: Cancel out common factors**

- The factor \( (x - 2) \) cancels out.
- The factor \( (x + 2) \) cancels out.

The simplified expression is:

\[
\frac{x(x - 9)}{(x + 5)(x - 3)}
\]

### Final Simplified Results:

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

5. \(\frac{x(x - 9)}{(x + 5)(x - 3)}\)

Would you like further details on these steps?

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Here are 5 related questions for further practice:

1. How do you factor the expression \(x^2 + 5x + 6\)?
2. What are the roots of the quadratic equation \(3x^2 + 11x - 4 = 0\)?
3. How do you simplify the expression \(\frac{(x^2 - 4)}{(x - 2)}\)?
4. What is the importance of canceling common factors in rational expressions?
5. How do you determine the domain of the expression \(\frac{1}{x+2}\)?

**Tip:** Always check for factors that can be simplified before performing any operations in algebraic expressions!

Explore how to solve complex rational expressions step-by-step with detailed explanations. This page covers algebraic concepts such as factoring quadratic equations and simplifying rational expressions. Ideal for high school students studying algebra and advanced mathematics.

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