1. **Problem Statement:** Simplify the expression $$\frac{2}{3x - 2y} - \frac{3y}{x(3x + 2y)} - \frac{8y}{9x^2 - 4y^2}$$ by performing the subtractions and reducing to the lowest terms. 2. **Identify the denominators and factor where possible:** - The denominators are $3x - 2y$, $x(3x + 2y)$, and $9x^2 - 4y^2$. - Note that $9x^2 - 4y^2$ is a difference of squares: $$9x^2 - 4y^2 = (3x - 2y)(3x + 2y)$$ 3. **Find the least common denominator (LCD):** - The LCD must include $x$, $3x - 2y$, and $3x + 2y$. - So, $$\text{LCD} = x(3x - 2y)(3x + 2y)$$ 4. **Rewrite each fraction with the LCD as denominator:** - First fraction: $$\frac{2}{3x - 2y} = \frac{2 \cdot x(3x + 2y)}{x(3x - 2y)(3x + 2y)} = \frac{2x(3x + 2y)}{x(3x - 2y)(3x + 2y)}$$ - Second fraction: $$- \frac{3y}{x(3x + 2y)} = - \frac{3y(3x - 2y)}{x(3x - 2y)(3x + 2y)}$$ - Third fraction: $$- \frac{8y}{(3x - 2y)(3x + 2y)} = - \frac{8y \cdot x}{x(3x - 2y)(3x + 2y)}$$ 5. **Combine the numerators over the common denominator:** $$\frac{2x(3x + 2y) - 3y(3x - 2y) - 8yx}{x(3x - 2y)(3x + 2y)}$$ 6. **Expand the numerators:** - $$2x(3x + 2y) = 6x^2 + 4xy$$ - $$-3y(3x - 2y) = -9xy + 6y^2$$ - $$-8yx = -8xy$$ 7. **Sum the numerator terms:** $$6x^2 + 4xy - 9xy + 6y^2 - 8xy = 6x^2 + (4xy - 9xy - 8xy) + 6y^2 = 6x^2 - 13xy + 6y^2$$ 8. **Final expression:** $$\frac{6x^2 - 13xy + 6y^2}{x(3x - 2y)(3x + 2y)}$$ 9. **Check if numerator factors:** - Try to factor $6x^2 - 13xy + 6y^2$. - Factors of $6x^2$ and $6y^2$ suggest trying $(2x - 3y)(3x - 2y)$: $$ (2x - 3y)(3x - 2y) = 6x^2 - 4xy - 9xy + 6y^2 = 6x^2 - 13xy + 6y^2$$ - Perfect match. 10. **Simplify by canceling common factors:** - The denominator has $(3x - 2y)$, numerator has $(3x - 2y)$. - Cancel $(3x - 2y)$: $$\frac{(2x - 3y)\cancel{(3x - 2y)}}{x \cancel{(3x - 2y)} (3x + 2y)} = \frac{2x - 3y}{x(3x + 2y)}$$ **Final answer:** $$\boxed{\frac{2x - 3y}{x(3x + 2y)}}$$