1. **Problem:** Simplify the algebraic fraction $$\frac{4}{3x-1} - \frac{3}{2x+1}$$ 2. **Formula and rules:** To subtract fractions, find a common denominator and combine the numerators: $$\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$$ 3. **Intermediate work:** $$\frac{4}{3x-1} - \frac{3}{2x+1} = \frac{4(2x+1) - 3(3x-1)}{(3x-1)(2x+1)}$$ 4. **Expand the numerators:** $$4(2x+1) = 8x + 4$$ $$3(3x-1) = 9x - 3$$ 5. **Substitute and simplify numerator:** $$8x + 4 - (9x - 3) = 8x + 4 - 9x + 3 = (8x - 9x) + (4 + 3) = -x + 7$$ 6. **Final simplified expression:** $$\frac{-x + 7}{(3x-1)(2x+1)}$$ 7. **Answer:** $$\boxed{\frac{7 - x}{(3x-1)(2x+1)}}$$