1. **State the problem:** We need to find the area of the shaded region, which is the area of the large square minus the area of the smaller square hole inside it. 2. **Write the formula for the area of a square:** $$\text{Area} = \text{side}^2$$ 3. **Calculate the area of the large square:** $$\text{Area}_{large} = (5z - 6)^2$$ 4. **Calculate the area of the smaller square:** $$\text{Area}_{small} = (z + 2)^2$$ 5. **Find the shaded area by subtracting the smaller square's area from the larger square's area:** $$\text{Area}_{shaded} = (5z - 6)^2 - (z + 2)^2$$ 6. **Expand both squares:** $$ (5z - 6)^2 = (5z)^2 - 2 \times 5z \times 6 + 6^2 = 25z^2 - 60z + 36 $$ $$ (z + 2)^2 = z^2 + 2 \times z \times 2 + 2^2 = z^2 + 4z + 4 $$ 7. **Subtract the two expressions:** $$ 25z^2 - 60z + 36 - (z^2 + 4z + 4) = 25z^2 - 60z + 36 - z^2 - 4z - 4 $$ 8. **Combine like terms:** $$ (25z^2 - z^2) + (-60z - 4z) + (36 - 4) = 24z^2 - 64z + 32 $$ 9. **Final answer:** The area of the shaded region is $$\boxed{24z^2 - 64z + 32}$$ square meters.