1. **State the problem:** Simplify the expression $$\frac{3\sqrt{6} - \sqrt{15}}{\sqrt{6} + 2\sqrt{3}}$$. 2. **Formula and rule:** To simplify a fraction with surds in the denominator, multiply numerator and denominator by the conjugate of the denominator to rationalize it. 3. **Identify the conjugate:** The conjugate of $$\sqrt{6} + 2\sqrt{3}$$ is $$\sqrt{6} - 2\sqrt{3}$$. 4. **Multiply numerator and denominator by the conjugate:** $$\frac{3\sqrt{6} - \sqrt{15}}{\sqrt{6} + 2\sqrt{3}} \times \frac{\sqrt{6} - 2\sqrt{3}}{\sqrt{6} - 2\sqrt{3}} = \frac{(3\sqrt{6} - \sqrt{15})(\sqrt{6} - 2\sqrt{3})}{(\sqrt{6} + 2\sqrt{3})(\sqrt{6} - 2\sqrt{3})}$$ 5. **Expand numerator:** $$3\sqrt{6} \times \sqrt{6} = 3 \times 6 = 18$$ $$3\sqrt{6} \times (-2\sqrt{3}) = -6\sqrt{18}$$ $$-\sqrt{15} \times \sqrt{6} = -\sqrt{90}$$ $$-\sqrt{15} \times (-2\sqrt{3}) = +2\sqrt{45}$$ So numerator is: $$18 - 6\sqrt{18} - \sqrt{90} + 2\sqrt{45}$$ 6. **Simplify radicals in numerator:** $$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$$ $$\sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}$$ $$\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$$ Substitute back: $$18 - 6 \times 3\sqrt{2} - 3\sqrt{10} + 2 \times 3\sqrt{5} = 18 - 18\sqrt{2} - 3\sqrt{10} + 6\sqrt{5}$$ 7. **Simplify denominator using difference of squares:** $$(\sqrt{6})^2 - (2\sqrt{3})^2 = 6 - 4 \times 3 = 6 - 12 = -6$$ 8. **Write the fraction:** $$\frac{18 - 18\sqrt{2} - 3\sqrt{10} + 6\sqrt{5}}{-6}$$ 9. **Divide each term by -6:** $$\frac{18}{-6} - \frac{18\sqrt{2}}{-6} - \frac{3\sqrt{10}}{-6} + \frac{6\sqrt{5}}{-6} = -3 + 3\sqrt{2} + \frac{1}{2}\sqrt{10} - \sqrt{5}$$ 10. **Final simplified expression:** $$-3 + 3\sqrt{2} + \frac{1}{2}\sqrt{10} - \sqrt{5}$$