1. **State the problem:** Simplify the expression $3\sqrt{7} + 1 + 2\sqrt{7}$. 2. **Identify like terms:** Terms involving $\sqrt{7}$ can be combined, and constants can be combined separately. 3. **Combine like terms:** $$3\sqrt{7} + 2\sqrt{7} = (3 + 2)\sqrt{7} = 5\sqrt{7}$$ 4. **Rewrite the expression:** $$5\sqrt{7} + 1$$ 5. **Final answer:** The simplified form is $5\sqrt{7} + 1$.