1. **Problem:** Divide the polynomial $x^2 + 5x + 6$ by $x + 2$ using long division. 2. **Formula and rules:** Polynomial long division is similar to numerical long division. We divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by this quotient, subtract from the dividend, and repeat with the remainder. 3. **Step 1:** Divide the leading term $x^2$ by $x$ to get $x$. 4. **Step 2:** Multiply the divisor $x + 2$ by $x$ to get $x^2 + 2x$. 5. **Step 3:** Subtract this from the dividend: $$ (x^2 + 5x + 6) - (x^2 + 2x) = \cancel{x^2} + 5x + 6 - \cancel{x^2} - 2x = 3x + 6 $$ 6. **Step 4:** Divide the new leading term $3x$ by $x$ to get $3$. 7. **Step 5:** Multiply the divisor $x + 2$ by $3$ to get $3x + 6$. 8. **Step 6:** Subtract this from the remainder: $$ (3x + 6) - (3x + 6) = \cancel{3x} + 6 - \cancel{3x} - 6 = 0 $$ 9. **Answer:** The quotient is $x + 3$ and the remainder is $0$, so $$ \frac{x^2 + 5x + 6}{x + 2} = x + 3 $$ This completes the division.