1. **Problem:** Evaluate the surface integral $$\iint_S \mathbf{F} \cdot \mathbf{n} \, ds$$ where $$\mathbf{F} = (3x - 2z)\mathbf{i} - (2x + y)\mathbf{j} + (y^2 + 2z)\mathbf{k}$$ and $$S$$ is the surface of the sphere centered at $$(1,2,3)$$ with radius $$4$$. 2. **Formula:** The Divergence Theorem states: $$\iint_S \mathbf{F} \cdot \mathbf{n} \, ds = \iiint_V \nabla \cdot \mathbf{F} \, dV$$ where $$V$$ is the volume enclosed by $$S$$. 3. **Calculate the divergence:** $$\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(3x - 2z) + \frac{\partial}{\partial y}(-2x - y) + \frac{\partial}{\partial z}(y^2 + 2z)$$ 4. Compute each partial derivative: $$\frac{\partial}{\partial x}(3x - 2z) = 3$$ $$\frac{\partial}{\partial y}(-2x - y) = -1$$ $$\frac{\partial}{\partial z}(y^2 + 2z) = 2$$ 5. Sum the partial derivatives: $$\nabla \cdot \mathbf{F} = 3 - 1 + 2 = 4$$ 6. Since divergence is constant, the volume integral becomes: $$\iiint_V 4 \, dV = 4 \times \text{Volume}(V)$$ 7. Volume of sphere with radius $$r=4$$ is: $$\text{Volume} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi 64 = \frac{256}{3} \pi$$ 8. Therefore, $$\iint_S \mathbf{F} \cdot \mathbf{n} \, ds = 4 \times \frac{256}{3} \pi = \frac{1024}{3} \pi$$ **Final answer:** $$\boxed{\frac{1024}{3} \pi}$$