1. **State the problem:** Simplify the expression $$3(x+1)(5x+3) - (2x+4)(6x-2)$$ and verify the expanded form. 2. **Recall the distributive property and FOIL method:** - To multiply binomials, use FOIL (First, Outer, Inner, Last). - Multiply each term carefully and combine like terms. 3. **Expand the first term:** $$3(x+1)(5x+3) = 3[(x)(5x) + (x)(3) + (1)(5x) + (1)(3)] = 3(5x^2 + 3x + 5x + 3) = 3(5x^2 + 8x + 3)$$ 4. **Distribute the 3:** $$3(5x^2 + 8x + 3) = 15x^2 + 24x + 9$$ 5. **Expand the second term:** $$(2x+4)(6x-2) = (2x)(6x) + (2x)(-2) + (4)(6x) + (4)(-2) = 12x^2 - 4x + 24x - 8$$ 6. **Combine like terms in the second term:** $$12x^2 + ( -4x + 24x ) - 8 = 12x^2 + 20x - 8$$ 7. **Subtract the second term from the first:** $$15x^2 + 24x + 9 - (12x^2 + 20x - 8)$$ 8. **Distribute the minus sign:** $$15x^2 + 24x + 9 - 12x^2 - 20x + 8$$ 9. **Combine like terms:** $$ (15x^2 - 12x^2) + (24x - 20x) + (9 + 8) = 3x^2 + 4x + 17$$ **Final answer:** $$3x^2 + 4x + 17$$