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 $x$: $$\frac{x^2}{x} = x$$ 4. **Step 2:** Multiply the divisor by $x$: $$x(x + 2) = x^2 + 2x$$ 5. **Step 3:** Subtract this from the dividend: $$\begin{aligned} (x^2 + 5x + 6) - (x^2 + 2x) &= x^2 + 5x + 6 - x^2 - 2x \\ &= (x^2 - x^2) + (5x - 2x) + 6 \\ &= 3x + 6 \end{aligned}$$ 6. **Step 4:** Divide the new leading term $3x$ by $x$: $$\frac{3x}{x} = 3$$ 7. **Step 5:** Multiply the divisor by $3$: $$3(x + 2) = 3x + 6$$ 8. **Step 6:** Subtract this from the remainder: $$\begin{aligned} (3x + 6) - (3x + 6) &= 3x + 6 - 3x - 6 \\ &= 0 \end{aligned}$$ 9. **Step 7:** Since the remainder is zero, the division is exact. 10. **Answer:** The quotient is $$x + 3$$ This means: $$\frac{x^2 + 5x + 6}{x + 2} = x + 3$$