To factor \( 9y^2 - 9y - 1 \), we need to find two binomials whose product gives the original expression. Here's the step-by-step process:

1. **Identify coefficients**: For the quadratic \( 9y^2 - 9y - 1 \), we have:
   - \( a = 9 \) (coefficient of \( y^2 \))
   - \( b = -9 \) (coefficient of \( y \))
   - \( c = -1 \) (constant term)

2. **Calculate the discriminant**: We use the discriminant \( b^2 - 4ac \) to check if the quadratic can be factored easily.
   \[
   \text{Discriminant} = (-9)^2 - 4(9)(-1) = 81 + 36 = 117
   \]
   The discriminant is not a perfect square, indicating that the expression cannot be factored over the rational numbers.

3. **Use the quadratic formula**: If exact factoring is not possible over rationals, we use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots:
   \[
   y = \frac{-(-9) \pm \sqrt{117}}{2(9)} = \frac{9 \pm \sqrt{117}}{18}
   \]
   The roots are:
   \[
   y_1 = \frac{9 + \sqrt{117}}{18}, \quad y_2 = \frac{9 - \sqrt{117}}{18}
   \]

Since exact factoring into rational numbers isn't possible, we recognize that \( 9y^2 - 9y - 1 \) is a prime polynomial over the rationals.

**Conclusion**: The quadratic \( 9y^2 - 9y - 1 \) cannot be factored into simpler rational expressions. However, it can be expressed as a product of two conjugate binomials involving irrational numbers if required.

Would you like details on factoring such polynomials with complex roots, or have any questions on this? 

Here are some related questions:

1. How do you factor a trinomial with rational coefficients?
2. What is the quadratic formula and how is it derived?
3. What is the significance of the discriminant in factoring polynomials?
4. How do we factor polynomials with complex roots?
5. Can we always factor polynomials using real numbers?

**Tip**: When factoring quadratics, always check the discriminant first. If it's not a perfect square, rational factoring isn't possible!

Explore how to factor the quadratic equation 9y^2 - 9y - 1 with step-by-step guidance. This problem delves into algebraic concepts like the discriminant and quadratic formula. Best suited for students in Grades 10-12.

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