1. **State the problem:** Simplify the expression $$(2x+3)(x+1)+(3+2x)(x-2)$$. 2. **Use the distributive property (FOIL) to expand each product:** $$(2x+3)(x+1) = 2x \cdot x + 2x \cdot 1 + 3 \cdot x + 3 \cdot 1 = 2x^2 + 2x + 3x + 3 = 2x^2 + 5x + 3$$ $$(3+2x)(x-2) = 3 \cdot x + 3 \cdot (-2) + 2x \cdot x + 2x \cdot (-2) = 3x - 6 + 2x^2 - 4x = 2x^2 - x - 6$$ 3. **Add the two expanded expressions:** $$ (2x^2 + 5x + 3) + (2x^2 - x - 6) = 2x^2 + 5x + 3 + 2x^2 - x - 6 $$ 4. **Combine like terms:** $$ 2x^2 + 2x^2 = 4x^2 $$ $$ 5x - x = 4x $$ $$ 3 - 6 = -3 $$ 5. **Final simplified expression:** $$ \boxed{4x^2 + 4x - 3} $$ This is the simplified form of the original expression.