1. **State the problem:** We want to complete the equation $$-2(z + 3) - z = \square - 4(z + 2)$$ so that it has exactly one solution for $z$. 2. **Rewrite and simplify the left side:** $$-2(z + 3) - z = -2z - 6 - z = -3z - 6$$ 3. **Rewrite the right side with the unknown expression $\square$:** $$\square - 4(z + 2) = \square - 4z - 8$$ 4. **Set the equation:** $$-3z - 6 = \square - 4z - 8$$ 5. **Rearrange to isolate $z$ terms:** $$-3z - 6 = \square - 4z - 8$$ Add $4z$ to both sides: $$-3z + 4z - 6 = \square - 8$$ Simplify: $$z - 6 = \square - 8$$ Add 8 to both sides: $$z + 2 = \square$$ 6. **Interpretation:** For the equation to have exactly one solution, the right side must be equal to $z + 2$. 7. **Check the options:** - $z + 2$ ✔ matches the derived expression. - $z$ ✘ does not match. - $2z + 3$ ✘ does not match. - $7$ ✘ does not match. - $z - 4$ ✘ does not match. - $-z$ ✘ does not match. **Final answer:** The only possible choice is $z + 2$. **Summary:** The equation $$-2(z + 3) - z = z + 2 - 4(z + 2)$$ has exactly one solution for $z$.