To multiply \( \sqrt{2}(12 + \sqrt{3}) \), we will distribute \( \sqrt{2} \) across the terms inside the parentheses.
\[ \sqrt{2}(12 + \sqrt{3}) = \sqrt{2} \cdot 12 + \sqrt{2} \cdot \sqrt{3} \]
Calculating each term:
- \( \sqrt{2} \cdot 12 = 12\sqrt{2} \)
- \( \sqrt{2} \cdot \sqrt{3} = \sqrt{2 \cdot 3} = \sqrt{6} \)
Now combine the terms:
\[ 12\sqrt{2} + \sqrt{6} \]
Thus, the simplified answer is:
\[ \boxed{12\sqrt{2} + \sqrt{6}} \]