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:** - Expand $$3(x+1)(5x+3)$$ as $$3[(x)(5x) + (x)(3) + (1)(5x) + (1)(3)] = 3(5x^2 + 3x + 5x + 3)$$. - Simplify inside the bracket: $$5x^2 + 8x + 3$$. - Multiply by 3: $$3 \times 5x^2 + 3 \times 8x + 3 \times 3 = 15x^2 + 24x + 9$$. 3. **Expand the second product:** - Expand $$(2x + 4)(6x - 2)$$ as $$(2x)(6x) + (2x)(-2) + (4)(6x) + (4)(-2) = 12x^2 - 4x + 24x - 8$$. - Combine like terms: $$12x^2 + 20x - 8$$. 4. **Rewrite the original expression with expanded terms:** $$15x^2 + 24x + 9 - (12x^2 + 20x - 8)$$. 5. **Distribute the minus sign:** $$15x^2 + 24x + 9 - 12x^2 - 20x + 8$$. 6. **Combine like terms:** - For $$x^2$$ terms: $$15x^2 - 12x^2 = 3x^2$$. - For $$x$$ terms: $$24x - 20x = 4x$$. - For constants: $$9 + 8 = 17$$. 7. **Final simplified expression:** $$3x^2 + 4x + 17$$. This is the simplified form of the given expression.