1. **State the problem:** We need to find the area of a square rug where each side measures $8x + 1$ feet. 2. **Formula for the area of a square:** The area $A$ of a square with side length $s$ is given by: $$A = s^2$$ 3. **Apply the formula:** Here, $s = 8x + 1$, so: $$A = (8x + 1)^2$$ 4. **Expand the square:** Use the formula for the square of a binomial: $$(a + b)^2 = a^2 + 2ab + b^2$$ So, $$A = (8x)^2 + 2 \times 8x \times 1 + 1^2 = 64x^2 + 16x + 1$$ 5. **Final answer:** The area of the square rug is: $$\boxed{64x^2 + 16x + 1}$$ square feet. This expression gives the area in terms of $x$.