1. **State the problem:** Simplify the expression $$\frac{(\sqrt{5}-2\sqrt{6})(\sqrt{3}+\sqrt{2})}{\sqrt{3}-\sqrt{2}}$$. 2. **Recall the formula and rules:** To simplify expressions with radicals, multiply out the numerators and denominators and rationalize if necessary. Use the distributive property and the identity $$a^2 - b^2 = (a-b)(a+b)$$ for difference of squares. 3. **Multiply the numerator:** $$(\sqrt{5}-2\sqrt{6})(\sqrt{3}+\sqrt{2}) = \sqrt{5}\cdot\sqrt{3} + \sqrt{5}\cdot\sqrt{2} - 2\sqrt{6}\cdot\sqrt{3} - 2\sqrt{6}\cdot\sqrt{2}$$ Calculate each term: $$\sqrt{5}\cdot\sqrt{3} = \sqrt{15}$$ $$\sqrt{5}\cdot\sqrt{2} = \sqrt{10}$$ $$2\sqrt{6}\cdot\sqrt{3} = 2\sqrt{18} = 2\times 3\sqrt{2} = 6\sqrt{2}$$ $$2\sqrt{6}\cdot\sqrt{2} = 2\sqrt{12} = 2\times 2\sqrt{3} = 4\sqrt{3}$$ So the numerator becomes: $$\sqrt{15} + \sqrt{10} - 6\sqrt{2} - 4\sqrt{3}$$ 4. **Rationalize the denominator:** Multiply numerator and denominator by the conjugate of the denominator $$\sqrt{3}+\sqrt{2}$$: $$\frac{\sqrt{15} + \sqrt{10} - 6\sqrt{2} - 4\sqrt{3}}{\sqrt{3}-\sqrt{2}} \times \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}} = \frac{(\sqrt{15} + \sqrt{10} - 6\sqrt{2} - 4\sqrt{3})(\sqrt{3}+\sqrt{2})}{(\sqrt{3})^2 - (\sqrt{2})^2}$$ 5. **Calculate the denominator:** $$(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1$$ 6. **Multiply the numerator terms:** Distribute each term: $$\sqrt{15}\cdot\sqrt{3} = \sqrt{45} = 3\sqrt{5}$$ $$\sqrt{15}\cdot\sqrt{2} = \sqrt{30}$$ $$\sqrt{10}\cdot\sqrt{3} = \sqrt{30}$$ $$\sqrt{10}\cdot\sqrt{2} = \sqrt{20} = 2\sqrt{5}$$ $$-6\sqrt{2}\cdot\sqrt{3} = -6\sqrt{6}$$ $$-6\sqrt{2}\cdot\sqrt{2} = -6\times 2 = -12$$ $$-4\sqrt{3}\cdot\sqrt{3} = -4\times 3 = -12$$ $$-4\sqrt{3}\cdot\sqrt{2} = -4\sqrt{6}$$ 7. **Combine like terms:** $$3\sqrt{5} + \sqrt{30} + \sqrt{30} + 2\sqrt{5} - 6\sqrt{6} - 12 - 12 - 4\sqrt{6}$$ Group similar terms: $$ (3\sqrt{5} + 2\sqrt{5}) + (\sqrt{30} + \sqrt{30}) + (-6\sqrt{6} - 4\sqrt{6}) + (-12 - 12)$$ Simplify: $$5\sqrt{5} + 2\sqrt{30} - 10\sqrt{6} - 24$$ 8. **Final answer:** $$\boxed{5\sqrt{5} + 2\sqrt{30} - 10\sqrt{6} - 24}$$