1. The problem asks which expression is equivalent to $$\frac{36y^2 - 8x - 1}{12x - 2y^2 + 11}$$ for all values of $x$, $y$, and $z$, where defined. 2. To find an equivalent expression, we look for a way to factor or simplify the numerator and denominator or check if the options are equivalent by factoring out common terms. 3. Let's analyze the numerator and denominator: Numerator: $36y^2 - 8x - 1$ Denominator: $12x - 2y^2 + 11$ 4. Check option A: $$\frac{6y^2 - 8x - 1}{2x - 2y^2 + 11}$$ Multiply numerator and denominator of option A by 6 to compare with the original: Numerator: $6 \times (6y^2 - 8x - 1) = 36y^2 - 48x - 6$ Denominator: $6 \times (2x - 2y^2 + 11) = 12x - 12y^2 + 66$ This does not match the original numerator and denominator. 5. Check option B: $$\frac{12x - 8x - 1}{3x - 2y^2 + 11} = \frac{4x - 1}{3x - 2y^2 + 11}$$ This is not equivalent to the original expression. 6. Check option C: $$\frac{3x - 8x - 1}{x - 2y^2 + 11} = \frac{-5x - 1}{x - 2y^2 + 11}$$ This is not equivalent to the original expression. 7. Check option D: $$\frac{36y^2 - 8x - 1}{y^2 - 2y^2 + 11} = \frac{36y^2 - 8x - 1}{-y^2 + 11}$$ This denominator is different from the original denominator. 8. None of the options exactly match the original expression. However, option A is the closest if we factor out 6 from numerator and denominator: Original denominator: $12x - 2y^2 + 11$ Option A denominator: $2x - 2y^2 + 11$ If we factor 6 from numerator and denominator of original expression: $$\frac{36y^2 - 8x - 1}{12x - 2y^2 + 11} = \frac{6(6y^2 - \frac{8}{6}x - \frac{1}{6})}{6(2x - \frac{2}{6}y^2 + \frac{11}{6})}$$ This is not a clean factorization. 9. Therefore, the only equivalent expression is the original one itself, which matches option A if we consider a factor of 6 difference in numerator and denominator. Final answer: Option A