1. Problem: Simplify $4\sqrt{6} \cdot \sqrt{15}$. 2. Use the property of radicals: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. 3. Multiply the constants and the radicals: $$4 \cdot \sqrt{6} \cdot \sqrt{15} = 4 \cdot \sqrt{6 \times 15} = 4 \sqrt{90}$$ 4. Simplify $\sqrt{90}$ if possible: $$\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3 \sqrt{10}$$ 5. Substitute back: $$4 \sqrt{90} = 4 \times 3 \sqrt{10} = 12 \sqrt{10}$$ 6. Final answer: $\boxed{12 \sqrt{10}}$