1. **State the problem:** We have a rectangular patio with dimensions length $8x - 10$ ft and width $2x$ ft. Inside it, there is a rectangular pool with length $6x - 14$ ft and width $2x$ ft. There is also a square hot tub with side length $x$ ft. We need to find: a) Area of the hot tub b) Area of the pool c) Area of the deck (patio area minus pool and hot tub areas) 2. **Formulas and rules:** - Area of a rectangle = length $\times$ width - Area of a square = side $\times$ side - To find the deck area, subtract the pool and hot tub areas from the patio area. 3. **Calculate the area of the hot tub:** $$\text{Area}_{hot\ tub} = x \times x = x^2$$ 4. **Calculate the area of the pool:** $$\text{Area}_{pool} = (6x - 14) \times 2x = 2x(6x - 14) = 12x^2 - 28x$$ 5. **Calculate the area of the patio:** $$\text{Area}_{patio} = (8x - 10) \times 2x = 2x(8x - 10) = 16x^2 - 20x$$ 6. **Calculate the area of the deck:** $$\text{Area}_{deck} = \text{Area}_{patio} - \text{Area}_{pool} - \text{Area}_{hot\ tub}$$ Substitute the expressions: $$= (16x^2 - 20x) - (12x^2 - 28x) - x^2$$ Simplify step-by-step: $$= 16x^2 - 20x - 12x^2 + 28x - x^2$$ $$= (16x^2 - 12x^2 - x^2) + (-20x + 28x)$$ $$= (3x^2) + (8x) = 3x^2 + 8x$$ **Final answers:** - Hot tub area: $x^2$ ft$^2$ - Pool area: $12x^2 - 28x$ ft$^2$ - Deck area: $3x^2 + 8x$ ft$^2$