Simplify \(\frac{1 - 2\sqrt{5}}{2 + 3\sqrt{2}}\).

A.

\(14(2\sqrt{2} + 6\sqrt{5} - 4\sqrt{10})\)

B.

\(\frac{1}{14}(2 - 3\sqrt{2} - 4\sqrt{5} - 6\sqrt{10})\)

C.

\(\frac{1}{14}(3\sqrt{2} + 4\sqrt{5} - 6\sqrt{10} - 2)\)

D.

\(14(2 + 3\sqrt{2} - 6\sqrt{5} + 4\sqrt{10})\)

Correct answer is C

\(\frac{1 - 2\sqrt{5}}{2 + 3\sqrt{2}} = (\frac{1 - 2\sqrt{5}}{2 + 3\sqrt{2}})(\frac{2 - 3\sqrt{2}}{2 - 3\sqrt{2}})\)

= \(\frac{2 - 3\sqrt{2} - 4\sqrt{5} + 6\sqrt{10}}{4 - 6\sqrt{2} + 6\sqrt{2} - 18}\)

= \(\frac{2 - 3\sqrt{2} - 4\sqrt{5} + 6\sqrt{10}}{-14}\)

= \(\frac{1}{14}(3\sqrt{2} + 4\sqrt{5} - 2 - 6\sqrt{10})\) (dividing through with the minus sign)