1. **State the problem:** We have the cubic function $y = ax^3 - 6x^2 + bx - 4$ and know it has a point of inflection at $(2, -2)$. We need to find the value of $a + b$. 2. **Recall the definition of a point of inflection:** A point of inflection occurs where the second derivative $y''$ is zero. 3. **Find the first derivative:** $$y' = \frac{d}{dx}(ax^3 - 6x^2 + bx - 4) = 3ax^2 - 12x + b$$ 4. **Find the second derivative:** $$y'' = \frac{d}{dx}(3ax^2 - 12x + b) = 6ax - 12$$ 5. **Use the inflection point condition:** At $x=2$, $y''=0$, so $$6a(2) - 12 = 0$$ $$12a - 12 = 0$$ $$12a = 12$$ $$a = 1$$ 6. **Use the point on the curve:** The point $(2, -2)$ lies on the graph, so $$y(2) = a(2)^3 - 6(2)^2 + b(2) - 4 = -2$$ Substitute $a=1$: $$1 \times 8 - 6 \times 4 + 2b - 4 = -2$$ $$8 - 24 + 2b - 4 = -2$$ $$-20 + 2b = -2$$ $$2b = 18$$ $$b = 9$$ 7. **Find $a + b$:** $$a + b = 1 + 9 = 10$$ **Final answer:** $a + b = 10$ which corresponds to option (A).