1. **State the problem:** Simplify the expression $$(5x+3)(2x-7) - (3x-2)(4x+1)$$. 2. **Use the distributive property (FOIL) to expand each product:** $$(5x+3)(2x-7) = 5x \cdot 2x + 5x \cdot (-7) + 3 \cdot 2x + 3 \cdot (-7) = 10x^2 - 35x + 6x - 21$$ $$(3x-2)(4x+1) = 3x \cdot 4x + 3x \cdot 1 - 2 \cdot 4x - 2 \cdot 1 = 12x^2 + 3x - 8x - 2$$ 3. **Simplify each expanded expression by combining like terms:** $$10x^2 - 35x + 6x - 21 = 10x^2 - 29x - 21$$ $$12x^2 + 3x - 8x - 2 = 12x^2 - 5x - 2$$ 4. **Substitute back into the original expression:** $$ (10x^2 - 29x - 21) - (12x^2 - 5x - 2) $$ 5. **Distribute the minus sign to the second group:** $$10x^2 - 29x - 21 - 12x^2 + 5x + 2$$ 6. **Combine like terms:** $$ (10x^2 - 12x^2) + (-29x + 5x) + (-21 + 2) = -2x^2 - 24x - 19$$ **Final answer:** $$\boxed{-2x^2 - 24x - 19}$$