1. **State the problem:** We have a rectangular prism with dimensions length $7$ cm, width $4$ cm, and height $4$ cm. A triangular pyramid (a tetrahedron) with vertical height $2$ cm is removed from the top front left corner. We need to find the volume of the remaining solid. 2. **Formula for volume of rectangular prism:** $$V_{prism} = \text{length} \times \text{width} \times \text{height}$$ 3. **Calculate volume of the prism:** $$V_{prism} = 7 \times 4 \times 4 = 112 \text{ cm}^3$$ 4. **Volume of the removed triangular pyramid:** The removed pyramid is a right triangular pyramid with base area and height given. The base is a right triangle with legs $4$ cm and $4$ cm (width and length of the removed corner), and the height of the pyramid is $2$ cm. 5. **Calculate base area of the triangular pyramid:** $$A_{base} = \frac{1}{2} \times 4 \times 4 = 8 \text{ cm}^2$$ 6. **Formula for volume of a pyramid:** $$V_{pyramid} = \frac{1}{3} \times A_{base} \times \text{height}$$ 7. **Calculate volume of the pyramid:** $$V_{pyramid} = \frac{1}{3} \times 8 \times 2 = \frac{16}{3} \approx 5.33 \text{ cm}^3$$ 8. **Calculate volume of the remaining solid:** $$V_{remaining} = V_{prism} - V_{pyramid} = 112 - \frac{16}{3} = \frac{336}{3} - \frac{16}{3} = \frac{320}{3} \approx 106.67 \text{ cm}^3$$ **Final answer:** The volume of the remaining solid is $$\boxed{\frac{320}{3} \text{ cm}^3 \approx 106.67 \text{ cm}^3}$$.