1. **State the problem:** We need to find the value of $q$ in the expression $$(\sqrt{3} - 5\sqrt{2})(\sqrt{3} + \sqrt{2}) = p + q\sqrt{6}.$$ 2. **Expand the product:** Use the distributive property (FOIL method): $$ (\sqrt{3})(\sqrt{3}) + (\sqrt{3})(\sqrt{2}) - (5\sqrt{2})(\sqrt{3}) - (5\sqrt{2})(\sqrt{2}) $$ 3. **Simplify each term:** - $(\sqrt{3})(\sqrt{3}) = 3$ - $(\sqrt{3})(\sqrt{2}) = \sqrt{6}$ - $(5\sqrt{2})(\sqrt{3}) = 5\sqrt{6}$ - $(5\sqrt{2})(\sqrt{2}) = 5 \times 2 = 10$ 4. **Combine the terms:** $$ 3 + \sqrt{6} - 5\sqrt{6} - 10 = (3 - 10) + (\sqrt{6} - 5\sqrt{6}) = -7 - 4\sqrt{6} $$ 5. **Identify $p$ and $q$:** From the expression $p + q\sqrt{6}$, we have $p = -7$ and $q = -4$. **Final answer:** $q = -4$ which corresponds to option (b).