1. **State the problem:** Simplify the expression $$\frac{x + 1}{2x + 9} - \frac{2x + 3}{4x - 1}$$. 2. **Find a common denominator:** The denominators are $2x + 9$ and $4x - 1$. The common denominator is their product: $$(2x + 9)(4x - 1)$$. 3. **Rewrite each fraction with the common denominator:** $$\frac{x + 1}{2x + 9} = \frac{(x + 1)(4x - 1)}{(2x + 9)(4x - 1)}$$ $$\frac{2x + 3}{4x - 1} = \frac{(2x + 3)(2x + 9)}{(4x - 1)(2x + 9)}$$ 4. **Subtract the numerators:** $$\frac{(x + 1)(4x - 1) - (2x + 3)(2x + 9)}{(2x + 9)(4x - 1)}$$ 5. **Expand the numerators:** $$(x + 1)(4x - 1) = 4x^2 - x + 4x - 1 = 4x^2 + 3x - 1$$ $$(2x + 3)(2x + 9) = 4x^2 + 18x + 6x + 27 = 4x^2 + 24x + 27$$ 6. **Subtract the expanded numerators:** $$4x^2 + 3x - 1 - (4x^2 + 24x + 27) = 4x^2 + 3x - 1 - 4x^2 - 24x - 27 = -21x - 28$$ 7. **Write the simplified expression:** $$\frac{-21x - 28}{(2x + 9)(4x - 1)}$$ 8. **Factor the numerator:** $$-21x - 28 = -7(3x + 4)$$ 9. **Final simplified form:** $$\frac{-7(3x + 4)}{(2x + 9)(4x - 1)}$$ This is the simplified expression for the given subtraction of fractions.