1. **State the problem:** We need to find the value of $k$ such that $(x + 2)$ is a factor of the polynomial $x^3 + 2x^2 + kx + 6$. 2. **Recall the Factor Theorem:** If $(x + 2)$ is a factor, then the polynomial evaluated at $x = -2$ must be zero. 3. **Apply the Factor Theorem:** Substitute $x = -2$ into the polynomial: $$(-2)^3 + 2(-2)^2 + k(-2) + 6 = 0$$ 4. **Simplify the expression:** $$-8 + 2(4) - 2k + 6 = 0$$ $$-8 + 8 - 2k + 6 = 0$$ $$6 - 2k = 0$$ 5. **Solve for $k$:** $$6 = 2k$$ $$k = 3$$ **Final answer:** $k = 3$