1. **State the problem:** Simplify the expression $2\sqrt{5} \cdot 3\sqrt{40}$.\n\n2. **Recall the multiplication rule for radicals:** $a\sqrt{b} \cdot c\sqrt{d} = (a \cdot c) \sqrt{b \cdot d}$.\n\n3. **Apply the rule:**\n$$2\sqrt{5} \cdot 3\sqrt{40} = (2 \cdot 3) \sqrt{5 \cdot 40} = 6 \sqrt{200}.$$\n\n4. **Simplify the radical:**\nFactor 200 into $100 \times 2$, since $100$ is a perfect square.\n$$6 \sqrt{200} = 6 \sqrt{100 \times 2} = 6 \sqrt{100} \sqrt{2} = 6 \times 10 \sqrt{2} = 60 \sqrt{2}.$$\n\n5. **Final answer:**\n$$2\sqrt{5} \cdot 3\sqrt{40} = 60 \sqrt{2}.$$