1. **Problem Statement:** We are given four rectangles with different lengths and widths. We need to write expressions for their perimeter and area, then simplify these expressions. 2. **Formulas:** - Perimeter of a rectangle: $$P = 2(\text{length} + \text{width})$$ - Area of a rectangle: $$A = \text{length} \times \text{width}$$ 3. **Top-left rectangle:** - Length = $2y$ - Width = $y - 2$ **Perimeter:** $$P = 2(2y + (y - 2)) = 2(2y + y - 2) = 2(3y - 2) = 6y - 4$$ **Area:** $$A = 2y \times (y - 2) = 2y(y - 2) = 2y^2 - 4y$$ 4. **Top-right rectangle:** - Length = $2x + 4$ - Width = $7$ **Perimeter:** $$P = 2((2x + 4) + 7) = 2(2x + 11) = 4x + 22$$ **Area:** $$A = (2x + 4) \times 7 = 14x + 28$$ 5. **Bottom-left rectangle:** - Length is unlabeled (assumed as $2c + 2x$ from the problem statement) - Width = $7y - 2x$ **Perimeter:** $$P = 2((2c + 2x) + (7y - 2x)) = 2(2c + 2x + 7y - 2x) = 2(2c + 7y) = 4c + 14y$$ **Area:** $$A = (2c + 2x)(7y - 2x)$$ Expanding: $$= 2c \times 7y - 2c \times 2x + 2x \times 7y - 2x \times 2x = 14cy - 4cx + 14xy - 4x^2$$ 6. **Bottom-right rectangle:** - Length = $2x + y$ - Width = $x + 2$ **Perimeter:** $$P = 2((2x + y) + (x + 2)) = 2(3x + y + 2) = 6x + 2y + 4$$ **Area:** $$A = (2x + y)(x + 2)$$ Expanding: $$= 2x \times x + 2x \times 2 + y \times x + y \times 2 = 2x^2 + 4x + xy + 2y$$ **Summary:** - Top-left rectangle: Perimeter = $6y - 4$, Area = $2y^2 - 4y$ - Top-right rectangle: Perimeter = $4x + 22$, Area = $14x + 28$ - Bottom-left rectangle: Perimeter = $4c + 14y$, Area = $14cy - 4cx + 14xy - 4x^2$ - Bottom-right rectangle: Perimeter = $6x + 2y + 4$, Area = $2x^2 + 4x + xy + 2y$ This step-by-step explanation shows how to use the perimeter and area formulas, substitute the given lengths and widths, and simplify the expressions clearly and simply.