1. **State the problem:** Factor the quadratic expression $$x^2 + 3\sqrt{3}x - 30$$. 2. **Recall the factoring formula:** For a quadratic $$ax^2 + bx + c$$, we look for two numbers that multiply to $$ac$$ and add to $$b$$. 3. Here, $$a = 1$$, $$b = 3\sqrt{3}$$, and $$c = -30$$. 4. Calculate $$ac = 1 \times (-30) = -30$$. 5. We need two numbers that multiply to $$-30$$ and add to $$3\sqrt{3}$$. 6. Consider the pair $$6$$ and $$-5$$: $$6 \times (-5) = -30$$ but $$6 + (-5) = 1$$, not $$3\sqrt{3}$$. 7. Try $$5\sqrt{3}$$ and $$-2\sqrt{3}$$: $$5\sqrt{3} \times (-2\sqrt{3}) = -10 \times 3 = -30$$ and $$5\sqrt{3} + (-2\sqrt{3}) = 3\sqrt{3}$$, which matches $$b$$. 8. Rewrite the middle term using these numbers: $$x^2 + 5\sqrt{3}x - 2\sqrt{3}x - 30$$. 9. Group terms: $$(x^2 + 5\sqrt{3}x) + (-2\sqrt{3}x - 30)$$. 10. Factor each group: $$x(x + 5\sqrt{3}) - 2\sqrt{3}(x + 5\sqrt{3})$$. 11. Factor out the common binomial: $$(x - 2\sqrt{3})(x + 5\sqrt{3})$$. **Final answer:** $$\boxed{(x - 2\sqrt{3})(x + 5\sqrt{3})}$$