1. **State the problem:** We are given a parallelepiped prism with a surface area of 504 cm². The dimensions include a height of 4 cm, a side length of 12 cm, and a slant edge length expressed as $3x + 20$. We need to form an equation for the surface area and solve for $x$. 2. **Identify the surface area formula:** The surface area $S$ of a parallelepiped prism is the sum of the areas of all its faces. Since it is a prism with rectangular faces, the surface area is: $$S = 2(lw + lh + wh)$$ where $l$, $w$, and $h$ are the lengths of the edges. 3. **Assign dimensions:** From the problem: - Height $h = 4$ cm - One side length $w = 12$ cm - The other side length $l = 3x + 20$ 4. **Write the surface area equation:** Substitute into the formula: $$504 = 2((3x + 20) \times 12 + (3x + 20) \times 4 + 12 \times 4)$$ 5. **Simplify inside the parentheses:** $$504 = 2(12(3x + 20) + 4(3x + 20) + 48)$$ $$= 2(36x + 240 + 12x + 80 + 48)$$ $$= 2(48x + 368)$$ 6. **Distribute the 2:** $$504 = 96x + 736$$ 7. **Solve for $x$:** $$96x = 504 - 736$$ $$96x = -232$$ $$x = \frac{-232}{96} = -\frac{29}{12} \approx -2.42$$ 8. **Interpretation:** The value of $x$ is approximately $-2.42$. This negative value may indicate a need to recheck the problem context or dimensions, but mathematically this is the solution. **Final answer:** $$x = -\frac{29}{12}$$