1. **State the problem:** Simplify the expression $$\frac{\sqrt{3} \left(3 + \sqrt{\frac{2}{3}}\right) \sqrt{3}}{\sqrt{3}}$$. 2. **Rewrite the expression:** The numerator is $$\sqrt{3} \times \left(3 + \sqrt{\frac{2}{3}}\right) \times \sqrt{3}$$ and the denominator is $$\sqrt{3}$$. 3. **Multiply the terms in the numerator:** Since $$\sqrt{3} \times \sqrt{3} = 3$$, the numerator becomes $$3 \times \left(3 + \sqrt{\frac{2}{3}}\right)$$. 4. **Write the expression now:** $$\frac{3 \left(3 + \sqrt{\frac{2}{3}}\right)}{\sqrt{3}}$$. 5. **Distribute the 3 in the numerator:** $$\frac{9 + 3 \sqrt{\frac{2}{3}}}{\sqrt{3}}$$. 6. **Simplify the term inside the square root:** $$\sqrt{\frac{2}{3}} = \frac{\sqrt{2}}{\sqrt{3}}$$. 7. **Substitute back:** $$\frac{9 + 3 \times \frac{\sqrt{2}}{\sqrt{3}}}{\sqrt{3}} = \frac{9 + \frac{3 \sqrt{2}}{\sqrt{3}}}{\sqrt{3}}$$. 8. **Combine the terms over a common denominator:** Write 9 as $$\frac{9 \sqrt{3}}{\sqrt{3}}$$ to have a common denominator $$\sqrt{3}$$: $$\frac{\frac{9 \sqrt{3}}{\sqrt{3}} + \frac{3 \sqrt{2}}{\sqrt{3}}}{\sqrt{3}} = \frac{\frac{9 \sqrt{3} + 3 \sqrt{2}}{\sqrt{3}}}{\sqrt{3}}$$. 9. **Simplify the complex fraction:** $$\frac{\frac{9 \sqrt{3} + 3 \sqrt{2}}{\sqrt{3}}}{\sqrt{3}} = \frac{9 \sqrt{3} + 3 \sqrt{2}}{\sqrt{3} \times \sqrt{3}} = \frac{9 \sqrt{3} + 3 \sqrt{2}}{3}$$. 10. **Factor out 3 in the numerator:** $$\frac{3 (3 \sqrt{3} + \sqrt{2})}{3}$$. 11. **Cancel the 3 in numerator and denominator:** $$\frac{\cancel{3} (3 \sqrt{3} + \sqrt{2})}{\cancel{3}} = 3 \sqrt{3} + \sqrt{2}$$. **Final answer:** $$3 \sqrt{3} + \sqrt{2}$$