1. **State the problem:** We need to match each quadratic expression with its correct factored form. 2. **Recall the difference of squares formula:** $$a^2 - b^2 = (a - b)(a + b)$$ This formula helps factor expressions that are the difference of two perfect squares. 3. **Analyze each quadratic expression:** - Expression 1: $45y^{2} - 80x^{2}$ - Factor out the greatest common factor (GCF): $$45y^{2} - 80x^{2} = 5(9y^{2} - 16x^{2})$$ - Recognize $9y^{2} = (3y)^2$ and $16x^{2} = (4x)^2$, so: $$5(9y^{2} - 16x^{2}) = 5((3y)^2 - (4x)^2) = 5(3y - 4x)(3y + 4x)$$ - Expression 2: $160x^{2} - 90y^{2}$ - Factor out GCF: $$160x^{2} - 90y^{2} = 10(16x^{2} - 9y^{2})$$ - Recognize $16x^{2} = (4x)^2$ and $9y^{2} = (3y)^2$, so: $$10((4x)^2 - (3y)^2) = 10(4x - 3y)(4x + 3y)$$ - Expression 3: $80y^{2} - 45x^{2}$ - Factor out GCF: $$80y^{2} - 45x^{2} = 5(16y^{2} - 9x^{2})$$ - Recognize $16y^{2} = (4y)^2$ and $9x^{2} = (3x)^2$, so: $$5((4y)^2 - (3x)^2) = 5(4y - 3x)(4y + 3x)$$ - Expression 4: $64x^{2} - 36y^{2}$ - Factor out GCF: $$64x^{2} - 36y^{2} = 4(16x^{2} - 9y^{2})$$ - Recognize $16x^{2} = (4x)^2$ and $9y^{2} = (3y)^2$, so: $$4((4x)^2 - (3y)^2) = 4(4x - 3y)(4x + 3y)$$ 4. **Match each expression with its factored form:** - $45y^{2} - 80x^{2} = 5(3y - 4x)(3y + 4x)$ - $160x^{2} - 90y^{2} = 10(4x - 3y)(4x + 3y)$ - $80y^{2} - 45x^{2} = 5(4y - 3x)(4y + 3x)$ - $64x^{2} - 36y^{2} = 4(4x - 3y)(4x + 3y)$ This completes the matching of quadratic expressions to their factored forms using the difference of squares formula.