1. **State the problem:** Divide the polynomial $x^3 + 2x^2 + 3x + 2$ by the binomial $x + 1$. 2. **Formula and rule:** Polynomial division can be done using long division or synthetic division. Here, we use long division. 3. **Set up the division:** Divide the leading term of the dividend $x^3$ by the leading term of the divisor $x$ to get $x^2$. 4. **Multiply and subtract:** Multiply $x^2$ by $x + 1$ to get $x^3 + x^2$. Subtract this from the original polynomial: $$ (x^3 + 2x^2 + 3x + 2) - (x^3 + x^2) = (2x^2 - x^2) + 3x + 2 = x^2 + 3x + 2 $$ 5. **Repeat the process:** Divide the new leading term $x^2$ by $x$ to get $x$. 6. **Multiply and subtract:** Multiply $x$ by $x + 1$ to get $x^2 + x$. Subtract: $$ (x^2 + 3x + 2) - (x^2 + x) = (3x - x) + 2 = 2x + 2 $$ 7. **Repeat again:** Divide $2x$ by $x$ to get $2$. 8. **Multiply and subtract:** Multiply $2$ by $x + 1$ to get $2x + 2$. Subtract: $$ (2x + 2) - (2x + 2) = 0 $$ 9. **Conclusion:** The quotient is $x^2 + x + 2$ and the remainder is $0$. **Final answer:** $$\frac{x^3 + 2x^2 + 3x + 2}{x + 1} = x^2 + x + 2$$