1. **Stating the problem:** We are given algebraic expressions and a floor plan with dimensions involving $x$. We need to simplify the algebraic fractions and understand the tile dimensions to help Stéphanie choose a tile format that meets the constraints. 2. **Simplify the first algebraic fraction:** $$\frac{14x^2 - 63x}{7x}$$ Factor numerator: $$14x^2 - 63x = 7x(2x - 9)$$ Rewrite fraction: $$\frac{\cancel{7x}(2x - 9)}{\cancel{7x}} = 2x - 9$$ 3. **Simplify the second algebraic fraction:** $$\frac{-10x^2 - 40x^2}{2x^2 - 8x} = \frac{-50x^2}{2x^2 - 8x}$$ Factor denominator: $$2x^2 - 8x = 2x(x - 4)$$ Rewrite fraction: $$\frac{-50x^2}{2x(x - 4)}$$ Cancel common factor $2x$: $$\frac{-\cancel{50}x^{\cancel{2}}}{\cancel{2}x(x - 4)} = \frac{-25x}{x - 4}$$ 4. **Tile dimensions and prices:** - Tuile A: Rectangle, length $x$, width 2, price 10 per tile. - Tuile B: Trapezoid, height $\frac{x}{3}$, bottom width 0.5, price 6 per tile, but not available. - Tuile C: Diamond, diagonals $x$ and 0.5, price 3.5 per tile. 5. **Floor plan dimensions:** - Central square side length: $x + 6$ - Other sides: $3x + 4$ and $9x$ 6. **Summary:** - Simplified algebraic fractions: $2x - 9$ and $\frac{-25x}{x - 4}$ - Tile B is unavailable. - Stéphanie can choose between Tuile A and Tuile C based on price and dimensions. **Final simplified expressions:** $$\frac{14x^2 - 63x}{7x} = 2x - 9$$ $$\frac{-10x^2 - 40x^2}{2x^2 - 8x} = \frac{-25x}{x - 4}$$