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!