1. **State the problem:** Simplify the expression $3(x + 1)(5x + 3) - (2x + 4)(6x - 2)$. 2. **Use the distributive property (FOIL) to expand each product:** $$3(x + 1)(5x + 3) = 3[(x)(5x) + (x)(3) + (1)(5x) + (1)(3)] = 3(5x^2 + 3x + 5x + 3) = 3(5x^2 + 8x + 3)$$ $$= 15x^2 + 24x + 9$$ Similarly, expand the second product: $$(2x + 4)(6x - 2) = (2x)(6x) + (2x)(-2) + (4)(6x) + (4)(-2) = 12x^2 - 4x + 24x - 8 = 12x^2 + 20x - 8$$ 3. **Substitute the expanded forms back into the expression:** $$3(x + 1)(5x + 3) - (2x + 4)(6x - 2) = (15x^2 + 24x + 9) - (12x^2 + 20x - 8)$$ 4. **Distribute the minus sign to the second group:** $$= 15x^2 + 24x + 9 - 12x^2 - 20x + 8$$ 5. **Combine like terms:** $$= (15x^2 - 12x^2) + (24x - 20x) + (9 + 8) = 3x^2 + 4x + 17$$ **Final answer:** $$\boxed{3x^2 + 4x + 17}$$