1. **Problem statement:** We have a rectangular prism with length $p$ cm, width and height each $2$ cm less than the length, i.e., width = height = $p-2$ cm. 2. **Part a) Surface area expression:** Surface area (SA) of a rectangular prism is given by: $$SA = 2(lw + lh + wh)$$ where $l = p$, $w = p-2$, and $h = p-2$. 3. **Calculate each term:** $$lw = p(p-2) = p^2 - 2p$$ $$lh = p(p-2) = p^2 - 2p$$ $$wh = (p-2)(p-2) = (p-2)^2 = p^2 - 4p + 4$$ 4. **Sum inside the parentheses:** $$lw + lh + wh = (p^2 - 2p) + (p^2 - 2p) + (p^2 - 4p + 4) = 3p^2 - 8p + 4$$ 5. **Surface area:** $$SA = 2(3p^2 - 8p + 4) = 6p^2 - 16p + 8$$ 6. **Factor the surface area expression:** Factor out 2: $$SA = 2(3p^2 - 8p + 4)$$ Try factoring the quadratic inside: $$3p^2 - 8p + 4 = (3p - 2)(p - 2)$$ 7. **Final factored form:** $$SA = 2(3p - 2)(p - 2)$$ --- 8. **Part b) Total length of edges of two square faces:** The two square faces are the ones with sides $p-2$ (width and height). Each square has 4 edges of length $p-2$, so two squares have: $$2 imes 4 imes (p-2) = 8(p-2)$$ --- 9. **Part c) Given ratio and find volume:** The ratio of surface area to total edge length of the two squares is $10:1$: $$\frac{SA}{8(p-2)} = 10$$ Substitute $SA$ from part a): $$\frac{2(3p - 2)(p - 2)}{8(p-2)} = 10$$ Simplify numerator and denominator: $$\frac{2(3p - 2)(p - 2)}{8(p-2)} = \frac{2(3p - 2)}{8} = \frac{3p - 2}{4}$$ Set equal to 10: $$\frac{3p - 2}{4} = 10$$ Multiply both sides by 4: $$3p - 2 = 40$$ Add 2: $$3p = 42$$ Divide by 3: $$p = 14$$ 10. **Calculate volume:** Volume $V = l imes w imes h = p imes (p-2) imes (p-2)$ Substitute $p=14$: $$V = 14 imes 12 imes 12 = 2016$$ **Final answer:** - Surface area: $$SA = 2(3p - 2)(p - 2)$$ - Total length of edges of two square faces: $$8(p-2)$$ - Volume when ratio is 10:1: $$2016$$ cubic cm