1. **State the problem:** Simplify and verify the expression $$(\sqrt{3} + \sqrt{2})^2 - (\sqrt{3} - \sqrt{2})^2 = 4 \sqrt{6}$$. 2. **Recall the formula:** Use the difference of squares formula: $$a^2 - b^2 = (a - b)(a + b)$$. 3. **Apply the formula:** Let $$a = \sqrt{3} + \sqrt{2}$$ and $$b = \sqrt{3} - \sqrt{2}$$. Then, $$ (\sqrt{3} + \sqrt{2})^2 - (\sqrt{3} - \sqrt{2})^2 = (a)^2 - (b)^2 = (a - b)(a + b) $$ 4. **Calculate $$a - b$$:** $$ a - b = (\sqrt{3} + \sqrt{2}) - (\sqrt{3} - \sqrt{2}) = \sqrt{3} + \sqrt{2} - \sqrt{3} + \sqrt{2} = 2 \sqrt{2} $$ 5. **Calculate $$a + b$$:** $$ a + b = (\sqrt{3} + \sqrt{2}) + (\sqrt{3} - \sqrt{2}) = \sqrt{3} + \sqrt{2} + \sqrt{3} - \sqrt{2} = 2 \sqrt{3} $$ 6. **Multiply $$a - b$$ and $$a + b$$:** $$ (2 \sqrt{2})(2 \sqrt{3}) = 4 \sqrt{6} $$ 7. **Conclusion:** The original expression simplifies exactly to $$4 \sqrt{6}$$, confirming the equality. **Final answer:** $$ (\sqrt{3} + \sqrt{2})^2 - (\sqrt{3} - \sqrt{2})^2 = 4 \sqrt{6} $$