1. **State the problem:** We need to find the value to replace the box (☐) in the equation $$7 + 3x - 1 = \boxed{\phantom{a}} (x + 2)$$ so that the equation has infinitely many solutions. 2. **Rewrite the equation:** Simplify the left side: $$7 + 3x - 1 = 6 + 3x$$ So the equation becomes: $$6 + 3x = \boxed{a} (x + 2)$$ where $a$ is the value in the box. 3. **Condition for infinitely many solutions:** For the equation to have infinitely many solutions, both sides must be identical expressions for all $x$. This means the coefficients of $x$ and the constant terms must be equal. 4. **Expand the right side:** $$\boxed{a} (x + 2) = a x + 2a$$ 5. **Set coefficients equal:** - Coefficient of $x$: $3 = a$ - Constant term: $6 = 2a$ 6. **Solve for $a$ from constant term:** $$6 = 2a \implies a = \frac{6}{2} = 3$$ 7. **Check if $a=3$ satisfies coefficient of $x$:** $$3 = 3$$ which is true. 8. **Conclusion:** The value in the box must be $3$ for the equation to have infinitely many solutions. **Final answer:** $3$ (option Ⓒ)