1. **State the problem:** Find the area of the shaded region, which is the area of the larger square minus the area of the smaller square cut out from inside it. 2. **Identify the dimensions:** - Larger square side length: $5y - 3$ - Smaller square side length: $y + 1$ 3. **Formula for area of a square:** $$\text{Area} = \text{side}^2$$ 4. **Calculate the area of the larger square:** $$ (5y - 3)^2 = (5y)^2 - 2 \times 5y \times 3 + 3^2 = 25y^2 - 30y + 9 $$ 5. **Calculate the area of the smaller square:** $$ (y + 1)^2 = y^2 + 2y + 1 $$ 6. **Find the shaded area by subtracting the smaller square area from the larger square area:** $$ \text{Shaded area} = (5y - 3)^2 - (y + 1)^2 $$ 7. **Substitute the expanded forms:** $$ 25y^2 - 30y + 9 - (y^2 + 2y + 1) $$ 8. **Simplify by distributing the minus sign:** $$ 25y^2 - 30y + 9 - y^2 - 2y - 1 $$ 9. **Combine like terms:** $$ (25y^2 - y^2) + (-30y - 2y) + (9 - 1) = 24y^2 - 32y + 8 $$ 10. **Check the given answer:** The problem states the area is $5y^2 + 2y - 3$, but our calculation shows $24y^2 - 32y + 8$. This suggests a possible error in the problem statement or interpretation. **Final answer:** $$\boxed{24y^2 - 32y + 8}$$ This is the simplified expression for the shaded area based on the given dimensions.