1. **Problem statement:** We have a photograph of dimensions 8 inches by 12 inches inside a frame of width $x$ inches around all sides. We want to find a function $A(x)$ that represents the area of the frame alone. 2. **Understanding the problem:** The total dimensions including the frame are increased by $x$ on each side, so the total width is $12 + 2x$ and the total height is $8 + 2x$. 3. **Formula for area of the frame alone:** The area of the frame alone is the area of the larger rectangle (photo + frame) minus the area of the photograph. $$A(x) = \text{Area of large rectangle} - \text{Area of photo}$$ 4. **Calculate areas:** - Area of large rectangle: $(12 + 2x)(8 + 2x)$ - Area of photo: $8 \times 12 = 96$ 5. **Write the function:** $$A(x) = (12 + 2x)(8 + 2x) - 96$$ 6. **Expand the product:** $$A(x) = 12 \times 8 + 12 \times 2x + 2x \times 8 + 2x \times 2x - 96$$ $$A(x) = 96 + 24x + 16x + 4x^2 - 96$$ 7. **Simplify:** $$A(x) = 4x^2 + 40x + \cancel{96} - \cancel{96} = 4x^2 + 40x$$ 8. **Final function:** $$\boxed{A(x) = 4x^2 + 40x}$$ This function gives the area of the frame alone in square inches for any frame width $x$.