1. **State the problem:** We need to find the polynomial representing the area of the shaded region between a large square with side length $z+6$ and a smaller square inside it with side length $z$. 2. **Formula for area of a square:** The area of a square is given by the formula: $$\text{Area} = \text{side}^2$$ 3. **Calculate the area of the large square:** $$\text{Area}_{large} = (z+6)^2$$ 4. **Calculate the area of the smaller square:** $$\text{Area}_{small} = z^2$$ 5. **Find the area of the shaded region:** This is the difference between the areas of the large and small squares: $$\text{Area}_{shaded} = (z+6)^2 - z^2$$ 6. **Expand the binomial:** $$ (z+6)^2 = z^2 + 2 \cdot z \cdot 6 + 6^2 = z^2 + 12z + 36 $$ 7. **Substitute and simplify:** $$\text{Area}_{shaded} = (z^2 + 12z + 36) - z^2$$ 8. **Cancel common terms:** $$\text{Area}_{shaded} = \cancel{z^2} + 12z + 36 - \cancel{z^2} = 12z + 36$$ 9. **Final answer:** The polynomial representing the area of the shaded region in descending powers of $z$ is: $$\boxed{12z + 36}$$