1. **Problem statement:** We need to find the total surface area of a trapezoidal prism box including the flaps. The flaps area is given as 84 cm². 2. **Given dimensions:** - Top base of trapezoid: 10 cm - Bottom base of trapezoid: 12 cm - Height of trapezoid: 8 cm - Length of prism: 24 cm - Rectangular side faces: 10 cm and 20 cm (these correspond to the non-parallel sides of the trapezoid) 3. **Step 1: Calculate the area of the trapezoidal base.** The area $A$ of a trapezoid is given by: $$A = \frac{(b_1 + b_2)}{2} \times h$$ where $b_1$ and $b_2$ are the lengths of the two bases, and $h$ is the height. Substitute the values: $$A = \frac{(10 + 12)}{2} \times 8 = \frac{22}{2} \times 8 = 11 \times 8 = 88 \text{ cm}^2$$ 4. **Step 2: Calculate the lateral surface area of the prism.** The lateral surface area is the perimeter of the trapezoid times the length of the prism. First, find the lengths of the non-parallel sides. Given as 10 cm and 20 cm. Calculate the perimeter $P$ of the trapezoid: $$P = 10 + 12 + 10 + 20 = 52 \text{ cm}$$ Then, lateral surface area $L$ is: $$L = P \times \text{length} = 52 \times 24 = 1248 \text{ cm}^2$$ 5. **Step 3: Calculate the total surface area of the box (without flaps).** The box has two trapezoidal bases and the lateral surface area: $$\text{Surface area} = 2 \times 88 + 1248 = 176 + 1248 = 1424 \text{ cm}^2$$ 6. **Step 4: Add the area of the flaps.** Total cardboard needed is the surface area plus the flaps area: $$\text{Total} = 1424 + 84 = 1508 \text{ cm}^2$$ **Final answer:** $$\boxed{1508 \text{ cm}^2}$$