1. **State the problem:** Simplify the expression $$\frac{5x^9}{8x^{11} + x^2} - \frac{15x^2}{8x^7 + x - 18}$$. 2. **Analyze each term:** The expression is a subtraction of two rational expressions with different denominators. 3. **Look for common factors or simplifications:** - The first denominator is $$8x^{11} + x^2$$. - The second denominator is $$8x^7 + x - 18$$. 4. **Check if denominators can be factored:** - For $$8x^{11} + x^2$$, factor out $$x^2$$: $$8x^{11} + x^2 = x^2(8x^9 + 1)$$. - For $$8x^7 + x - 18$$, no obvious factorization without complex methods. 5. **Since denominators are different and cannot be easily factored to a common denominator, the expression cannot be combined further in a simple way.** 6. **Final simplified form:** $$\frac{5x^9}{8x^{11} + x^2} - \frac{15x^2}{8x^7 + x - 18}$$ This is the simplest form without further factorization or common denominator. **Answer:** $$\frac{5x^9}{8x^{11} + x^2} - \frac{15x^2}{8x^7 + x - 18}$$