1. **State the problem:** Simplify the expression $$\frac{\sqrt{5} - 3\sqrt{3}}{2\sqrt{5} + 2\sqrt{3}}$$ by rationalizing the denominator. 2. **Formula and rule:** To rationalize a denominator with surds, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$2\sqrt{5} + 2\sqrt{3}$$ is $$2\sqrt{5} - 2\sqrt{3}$$. 3. **Multiply numerator and denominator by the conjugate:** $$\frac{\sqrt{5} - 3\sqrt{3}}{2\sqrt{5} + 2\sqrt{3}} \times \frac{2\sqrt{5} - 2\sqrt{3}}{2\sqrt{5} - 2\sqrt{3}} = \frac{(\sqrt{5} - 3\sqrt{3})(2\sqrt{5} - 2\sqrt{3})}{(2\sqrt{5} + 2\sqrt{3})(2\sqrt{5} - 2\sqrt{3})}$$ 4. **Expand numerator:** $$= \frac{2\sqrt{5} \times \sqrt{5} - 2\sqrt{5} \times 3\sqrt{3} - 2\sqrt{3} \times \sqrt{5} + 2\sqrt{3} \times 3\sqrt{3}}{(2\sqrt{5} + 2\sqrt{3})(2\sqrt{5} - 2\sqrt{3})}$$ 5. **Simplify terms in numerator:** $$= \frac{2 \times 5 - 6 \times \sqrt{15} - 2 \times \sqrt{15} + 6 \times 3}{(2\sqrt{5} + 2\sqrt{3})(2\sqrt{5} - 2\sqrt{3})} = \frac{10 - 8\sqrt{15} + 18}{(2\sqrt{5} + 2\sqrt{3})(2\sqrt{5} - 2\sqrt{3})}$$ 6. **Combine like terms in numerator:** $$= \frac{28 - 8\sqrt{15}}{(2\sqrt{5} + 2\sqrt{3})(2\sqrt{5} - 2\sqrt{3})}$$ 7. **Expand denominator using difference of squares:** $$= \frac{28 - 8\sqrt{15}}{(2\sqrt{5})^2 - (2\sqrt{3})^2} = \frac{28 - 8\sqrt{15}}{4 \times 5 - 4 \times 3} = \frac{28 - 8\sqrt{15}}{20 - 12}$$ 8. **Simplify denominator:** $$= \frac{28 - 8\sqrt{15}}{8}$$ 9. **Simplify fraction by dividing numerator and denominator by 4:** $$= \frac{\cancel{4} \times 7 - \cancel{4} \times 2\sqrt{15}}{\cancel{4} \times 2} = \frac{7 - 2\sqrt{15}}{2}$$ **Final answer:** $$\boxed{\frac{7 - 2\sqrt{15}}{2}}$$