1. **Stating the problem:** Calculate the surface area of a triangular prism given the formula $$SA = (s_1 + s_2 + s_3)h + bh$$ where $s_1, s_2, s_3$ are the sides of the triangular base, $h$ is the height (length) of the prism, and $b$ and $h$ in the second term represent the base and height of the triangular face respectively. 2. **Understanding the formula:** The surface area of a triangular prism consists of the sum of the areas of the three rectangular side faces plus the areas of the two triangular bases. - The term $(s_1 + s_2 + s_3)h$ calculates the total area of the three rectangular faces, where each rectangle's area is a side length times the prism's height. - The term $bh$ calculates the area of one triangular base, so multiplying by 2 (implied in the formula) accounts for both triangular bases. 3. **Important rules:** - The perimeter of the triangular base is $P = s_1 + s_2 + s_3$. - The area of the triangular base is $A_{triangle} = \frac{1}{2}bh$. 4. **Rewriting the formula for clarity:** $$SA = Ph + 2 \times \frac{1}{2}bh = Ph + bh$$ 5. **Example calculation:** Suppose the triangular base sides are $s_1=3$, $s_2=4$, $s_3=5$, the base $b=4$, the height of the triangle $h=3$, and the prism height (length) $h=10$ (to avoid confusion, denote prism height as $H=10$). 6. **Calculate perimeter:** $$P = 3 + 4 + 5 = 12$$ 7. **Calculate area of triangular base:** $$A_{triangle} = \frac{1}{2} \times 4 \times 3 = 6$$ 8. **Calculate surface area:** $$SA = P \times H + 2 \times A_{triangle} = 12 \times 10 + 2 \times 6 = 120 + 12 = 132$$ 9. **Final answer:** The surface area of the triangular prism is $132$ square units.