1. **Problem Statement:** Find the area of the shaded region in a trapezoid with an inscribed rectangle. The trapezoid has bases $b_1 = 10$ in and $b_2 = 15$ in, and height $h = 8$ in. The inscribed rectangle has base $7$ in and height $5$ in. 2. **Formula for the area of a trapezoid:** $$A_{trapezoid} = \frac{1}{2} (b_1 + b_2) h$$ 3. **Calculate the trapezoid area:** $$A_{trapezoid} = \frac{1}{2} (10 + 15) \times 8 = \frac{1}{2} \times 25 \times 8 = 100$$ 4. **Formula for the area of a rectangle:** $$A_{rectangle} = base \times height$$ 5. **Calculate the rectangle area:** $$A_{rectangle} = 7 \times 5 = 35$$ 6. **Calculate the shaded area:** The shaded area is the trapezoid area minus the rectangle area: $$A_{shaded} = A_{trapezoid} - A_{rectangle} = 100 - 35 = 65$$ **Final answer:** $$\boxed{65 \text{ square inches}}$$