Let's simplify the expression \(\sqrt{5} \times (8 + 3\sqrt{6})\).
We can distribute \(\sqrt{5}\) across the terms in the parentheses:
\[ \sqrt{5} \times (8 + 3\sqrt{6}) = \sqrt{5} \times 8 + \sqrt{5} \times 3\sqrt{6} \]
This simplifies to:
\[ 8\sqrt{5} + 3\sqrt{5} \cdot \sqrt{6} \]
Now, we can simplify \(3\sqrt{5} \cdot \sqrt{6}\) using the property that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\):
\[ 3\sqrt{5} \cdot \sqrt{6} = 3\sqrt{30} \]
Therefore, the expression simplifies to:
\[ 8\sqrt{5} + 3\sqrt{30} \]
Now, looking at the answer choices provided:
a. \(8\sqrt{5} + 3\sqrt{30}\)
b. \(5\sqrt{10}\)
c. 130
d. \(\sqrt{40} + 15\sqrt{6}\)
The simplified result \(8\sqrt{5} + 3\sqrt{30}\) matches with option a.
So, the correct answer is:
a \(8\sqrt{5} + 3\sqrt{30}\).