1. **State the problem:** We have two rectangles: an outer rectangle with sides $5x + 8$ and $6x + 2$, and an inner rectangle with sides $x + 5$ and $3x$. The shaded region is the area of the outer rectangle minus the area of the inner rectangle. 2. **Write the formula for the area of a rectangle:** $$\text{Area} = \text{length} \times \text{width}$$ 3. **Calculate the area of the outer rectangle:** $$A_{outer} = (5x + 8)(6x + 2)$$ Multiply using distributive property: $$= 5x \times 6x + 5x \times 2 + 8 \times 6x + 8 \times 2$$ $$= 30x^2 + 10x + 48x + 16$$ $$= 30x^2 + 58x + 16$$ 4. **Calculate the area of the inner rectangle:** $$A_{inner} = (x + 5)(3x)$$ Multiply: $$= x \times 3x + 5 \times 3x$$ $$= 3x^2 + 15x$$ 5. **Find the shaded area by subtracting inner area from outer area:** $$A_{shaded} = A_{outer} - A_{inner}$$ $$= (30x^2 + 58x + 16) - (3x^2 + 15x)$$ $$= 30x^2 + 58x + 16 - 3x^2 - 15x$$ $$= (30x^2 - 3x^2) + (58x - 15x) + 16$$ $$= 27x^2 + 43x + 16$$ 6. **Final answer:** The polynomial representing the shaded area is $$\boxed{27x^2 + 43x + 16}$$