1. **State the problem:** Simplify the expression $f = (1 - \sqrt{3})(2 + \sqrt{6})(5 + \sqrt{2})$. 2. **Recall the distributive property:** To simplify a product of multiple binomials, multiply two at a time, then multiply the result by the next. 3. **Multiply the first two binomials:** $$ (1 - \sqrt{3})(2 + \sqrt{6}) = 1 \cdot 2 + 1 \cdot \sqrt{6} - \sqrt{3} \cdot 2 - \sqrt{3} \cdot \sqrt{6} $$ Simplify each term: $$ = 2 + \sqrt{6} - 2\sqrt{3} - \sqrt{18} $$ Since $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$, we have: $$ = 2 + \sqrt{6} - 2\sqrt{3} - 3\sqrt{2} $$ 4. **Now multiply this result by the third binomial $(5 + \sqrt{2})$:** $$ (2 + \sqrt{6} - 2\sqrt{3} - 3\sqrt{2})(5 + \sqrt{2}) $$ Multiply each term in the first polynomial by each term in the second: $$ = 2 \cdot 5 + 2 \cdot \sqrt{2} + \sqrt{6} \cdot 5 + \sqrt{6} \cdot \sqrt{2} - 2\sqrt{3} \cdot 5 - 2\sqrt{3} \cdot \sqrt{2} - 3\sqrt{2} \cdot 5 - 3\sqrt{2} \cdot \sqrt{2} $$ Simplify each term: $$ = 10 + 2\sqrt{2} + 5\sqrt{6} + \sqrt{12} - 10\sqrt{3} - 2\sqrt{6} - 15\sqrt{2} - 3 \times 2 $$ Since $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ and $\sqrt{2} \cdot \sqrt{2} = 2$, we get: $$ = 10 + 2\sqrt{2} + 5\sqrt{6} + 2\sqrt{3} - 10\sqrt{3} - 2\sqrt{6} - 15\sqrt{2} - 6 $$ 5. **Combine like terms:** - Constants: $10 - 6 = 4$ - $\sqrt{2}$ terms: $2\sqrt{2} - 15\sqrt{2} = -13\sqrt{2}$ - $\sqrt{6}$ terms: $5\sqrt{6} - 2\sqrt{6} = 3\sqrt{6}$ - $\sqrt{3}$ terms: $2\sqrt{3} - 10\sqrt{3} = -8\sqrt{3}$ 6. **Final simplified expression:** $$ 4 - 13\sqrt{2} + 3\sqrt{6} - 8\sqrt{3} $$