1. **State the problem:** Simplify the expression $5\sqrt{45} - 11\sqrt{20} + \sqrt{12} - \sqrt{75}$.\n\n2. **Recall the rule:** $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ and simplify square roots by factoring out perfect squares.\n\n3. **Simplify each term:**\n- $5\sqrt{45} = 5\sqrt{9 \times 5} = 5 \times 3 \sqrt{5} = 15\sqrt{5}$\n- $-11\sqrt{20} = -11\sqrt{4 \times 5} = -11 \times 2 \sqrt{5} = -22\sqrt{5}$\n- $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$\n- $-\sqrt{75} = -\sqrt{25 \times 3} = -5\sqrt{3}$\n\n4. **Rewrite the expression:** $15\sqrt{5} - 22\sqrt{5} + 2\sqrt{3} - 5\sqrt{3}$.\n\n5. **Combine like terms:**\n- For $\sqrt{5}$ terms: $15\sqrt{5} - 22\sqrt{5} = (15 - 22)\sqrt{5} = -7\sqrt{5}$\n- For $\sqrt{3}$ terms: $2\sqrt{3} - 5\sqrt{3} = (2 - 5)\sqrt{3} = -3\sqrt{3}$\n\n6. **Final simplified expression:** $-7\sqrt{5} - 3\sqrt{3}$