1. **State the problem:** Expand and simplify the expression $$(2 - \sqrt{5})(1 - 3\sqrt{5})$$. 2. **Formula used:** Use the distributive property (FOIL method) to expand the product of two binomials: $$(a + b)(c + d) = ac + ad + bc + bd$$. 3. **Apply the formula:** $$ (2 - \sqrt{5})(1 - 3\sqrt{5}) = 2 \times 1 + 2 \times (-3\sqrt{5}) + (-\sqrt{5}) \times 1 + (-\sqrt{5}) \times (-3\sqrt{5}) $$ 4. **Calculate each term:** - $2 \times 1 = 2$ - $2 \times (-3\sqrt{5}) = -6\sqrt{5}$ - $(-\sqrt{5}) \times 1 = -\sqrt{5}$ - $(-\sqrt{5}) \times (-3\sqrt{5}) = 3 \times 5 = 15$ (since $\sqrt{5} \times \sqrt{5} = 5$ and negative times negative is positive) 5. **Combine all terms:** $$ 2 - 6\sqrt{5} - \sqrt{5} + 15 $$ 6. **Simplify like terms:** $$ (2 + 15) + (-6\sqrt{5} - \sqrt{5}) = 17 - 7\sqrt{5} $$ **Final answer:** $$17 - 7\sqrt{5}$$