1. **State the problem:** Simplify the expression $$\frac{15x^2}{x^2 + 7x - 18} - \frac{6x^5}{x^2 - 11x + 18}$$ and analyze its components. 2. **Factor the denominators:** - Factor $$x^2 + 7x - 18$$: $$x^2 + 7x - 18 = (x + 9)(x - 2)$$ - Factor $$x^2 - 11x + 18$$: $$x^2 - 11x + 18 = (x - 9)(x - 2)$$ 3. **Rewrite the expression with factored denominators:** $$\frac{15x^2}{(x + 9)(x - 2)} - \frac{6x^5}{(x - 9)(x - 2)}$$ 4. **Find the common denominator:** The common denominator is $$ (x + 9)(x - 2)(x - 9) $$. 5. **Rewrite each fraction with the common denominator:** $$\frac{15x^2 (x - 9)}{(x + 9)(x - 2)(x - 9)} - \frac{6x^5 (x + 9)}{(x - 9)(x - 2)(x + 9)}$$ 6. **Expand the numerators:** - First numerator: $$15x^2 (x - 9) = 15x^3 - 135x^2$$ - Second numerator: $$6x^5 (x + 9) = 6x^6 + 54x^5$$ 7. **Combine the fractions:** $$\frac{15x^3 - 135x^2 - (6x^6 + 54x^5)}{(x + 9)(x - 2)(x - 9)} = \frac{15x^3 - 135x^2 - 6x^6 - 54x^5}{(x + 9)(x - 2)(x - 9)}$$ 8. **Simplify the numerator:** $$-6x^6 - 54x^5 + 15x^3 - 135x^2$$ 9. **Factor out the greatest common factor (GCF) from the numerator:** GCF is $$-3x^2$$: $$-3x^2 (2x^4 + 18x^3 - 5x + 45)$$ 10. **Final simplified expression:** $$\frac{-3x^2 (2x^4 + 18x^3 - 5x + 45)}{(x + 9)(x - 2)(x - 9)}$$ **Note:** The function has vertical asymptotes at the roots of the denominators: $$x = -9, 2, 9$$. This completes the simplification and analysis of the given rational expression.