1. **State the problem:** Add the terms $4\sqrt{5}$ and $3\sqrt{5}$.\n\n2. **Identify like terms:** Both terms have the same radical part $\sqrt{5}$, so they are like terms and can be added by adding their coefficients.\n\n3. **Add the coefficients:** $4 + 3 = 7$.\n\n4. **Write the result:** Combine the sum of coefficients with the common radical part: $$7\sqrt{5}$$\n\n**Final answer:** $7\sqrt{5}$