1. **State the problem:** We need to find the surface area of a triangular prism with a triangular base of base $8$ cm and height $15$ cm, a slant edge of $17$ cm, and prism length $14$ cm. 2. **Formula for surface area of a triangular prism:** $$\text{Surface Area} = 2 \times \text{Area of triangular base} + \text{Perimeter of base} \times \text{Length}$$ 3. **Calculate the area of the triangular base:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 15 = 60 \text{ cm}^2$$ 4. **Find the perimeter of the triangular base:** The sides are $8$ cm, $15$ cm (height), and $17$ cm (slant edge). So, $$\text{Perimeter} = 8 + 15 + 17 = 40 \text{ cm}$$ 5. **Calculate the lateral surface area:** $$\text{Lateral Surface Area} = \text{Perimeter} \times \text{Length} = 40 \times 14 = 560 \text{ cm}^2$$ 6. **Calculate total surface area:** $$\text{Surface Area} = 2 \times 60 + 560 = 120 + 560 = 680 \text{ cm}^2$$ **Final answer:** The surface area of the triangular prism is $680$ cm$^2$.