1. **State the problem:** Simplify the expression $$\frac{6}{x^2 - 3x - 28} - \frac{2}{2x^2 + 3x - 20}$$ to the form $$\frac{2(ax - b)}{(x - c)(x + d)(2x - 5)}$$ where $a, b, c, d$ are positive integers. 2. **Factor the denominators:** - Factor $x^2 - 3x - 28$: $$x^2 - 3x - 28 = (x - 7)(x + 4)$$ - Factor $2x^2 + 3x - 20$: $$2x^2 + 3x - 20 = (2x - 5)(x + 4)$$ 3. **Rewrite the expression with factored denominators:** $$\frac{6}{(x - 7)(x + 4)} - \frac{2}{(2x - 5)(x + 4)}$$ 4. **Find common denominator:** The common denominator is $(x - 7)(x + 4)(2x - 5)$. 5. **Rewrite each fraction with the common denominator:** $$\frac{6(2x - 5)}{(x - 7)(x + 4)(2x - 5)} - \frac{2(x - 7)}{(x - 7)(x + 4)(2x - 5)}$$ 6. **Combine the numerators:** $$\frac{6(2x - 5) - 2(x - 7)}{(x - 7)(x + 4)(2x - 5)} = \frac{12x - 30 - 2x + 14}{(x - 7)(x + 4)(2x - 5)} = \frac{10x - 16}{(x - 7)(x + 4)(2x - 5)}$$ 7. **Factor numerator if possible:** $$10x - 16 = 2(5x - 8)$$ 8. **Final simplified form:** $$\frac{2(5x - 8)}{(x - 7)(x + 4)(2x - 5)}$$ 9. **Compare with given form:** $$2(ax - b) = 2(5x - 8) \Rightarrow a = 5, b = 8$$ Given $c = 7$, $d = 4$ match the factors. **Note:** The problem states $a=41$, $b=29$, $c=7$, $d=4$ but the simplification yields $a=5$, $b=8$. Possibly a typo in the problem. --- **Answer:** $$\boxed{\frac{2(5x - 8)}{(x - 7)(x + 4)(2x - 5)}}$$