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The given expression is:

\[
\frac{9x^2 - 15x - 6}{8x^2 - 9}
\]

### Simplification Steps:

1. **Factor the numerator** \(9x^2 - 15x - 6\):
   - First, factor out the greatest common factor (GCF), which is 3:
     \[
     9x^2 - 15x - 6 = 3(3x^2 - 5x - 2)
     \]
   - Now, factor the quadratic expression inside the parentheses: \(3x^2 - 5x - 2\). The factors are:
     \[
     3x^2 - 5x - 2 = (3x + 1)(x - 2)
     \]
   - Thus, the numerator becomes:
     \[
     9x^2 - 15x - 6 = 3(3x + 1)(x - 2)
     \]

2. **Factor the denominator** \(8x^2 - 9\):
   - This is a difference of squares:
     \[
     8x^2 - 9 = (2\sqrt{2}x + 3)(2\sqrt{2}x - 3)
     \]

3. **Final simplified expression**:
   \[
   \frac{3(3x + 1)(x - 2)}{(2\sqrt{2}x + 3)(2\sqrt{2}x - 3)}
   \]

Since there are no common factors between the numerator and denominator, this is the simplified form.

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Would you like to go deeper into the factoring steps or explore related concepts?

Here are 5 related questions you might find helpful:
1. How do you solve quadratic equations by factoring?
2. What are the rules for factoring the difference of squares?
3. Can this expression be evaluated for specific values of \( x \)?
4. How would you simplify similar rational expressions?
5. What happens if the numerator and denominator share common factors?

**Tip:** Always check for common factors before diving into complex factorizations!

Simplify the expression: (9x^2 - 15x - 6) / (8x^2 - 9).

Learn how to simplify the quadratic expression (9x^2 - 15x - 6) / (8x^2 - 9). This step-by-step solution involves factoring quadratic equations and applying the difference of squares rule. Suitable for students in Grades 9-12.

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