1. **State the problem:** We are given the volume of a box as $$V = x^3 + 3x^2 + 2x$$. We need to: a) Determine the possible dimensions of the box. b) Find the dimensions and volume when $$x = 20$$. 2. **Factor the volume expression:** To find the dimensions, we factor the polynomial volume expression. $$V = x^3 + 3x^2 + 2x$$ First, factor out the common factor $$x$$: $$V = x(x^2 + 3x + 2)$$ Next, factor the quadratic: $$x^2 + 3x + 2 = (x + 1)(x + 2)$$ So the volume factors as: $$V = x(x + 1)(x + 2)$$ 3. **Interpret the factors as dimensions:** The dimensions of the box are: - Length = $$x$$ - Width = $$x + 1$$ - Height = $$x + 2$$ 4. **Calculate dimensions and volume for $$x = 20$$:** - Length = $$20$$ - Width = $$20 + 1 = 21$$ - Height = $$20 + 2 = 22$$ Calculate volume: $$V = 20 \times 21 \times 22$$ Calculate step-by-step: $$20 \times 21 = 420$$ $$420 \times 22 = 9240$$ So, the volume is $$9240$$. **Final answers:** a) Dimensions: $$x$$, $$x+1$$, $$x+2$$ b) For $$x=20$$, dimensions are 20, 21, 22 and volume is 9240.