1) \( 8 x^{2}+18 x y-5 y^{2} \)

To factor the quadratic expression \( 8x^{2} + 18xy - 5y^{2} \), we'll look for two binomials of the form \((ax + by)(cx + dy)\) that multiply to give the original expression. ### Step 1: Identify Coefficients We need to find integers \(a\), \(b\), \(c\), and \(d\) such that: \[ (ac) = 8 \quad \text{(coefficient of } x^2\text{)} \] \[ (bd) = -5 \quad \text{(constant term)} \] \[ (ad + bc) = 18 \quad \text{(coefficient of } xy\text{)} \] ### Step 2: List Possible Pairs - **For \(ac = 8\):** Possible pairs \((a, c)\) are \((1, 8)\), \((2, 4)\), \((4, 2)\), \((8, 1)\). - **For \(bd = -5\):** Possible pairs \((b, d)\) are \((1, -5)\), \((-1, 5)\), \((5, -1)\), \((-5, 1)\). ### Step 3: Find the Correct Combination Let's test the pair \((a, c) = (2, 4)\) and \((b, d) = (5, -1)\): \[ ad + bc = (2)(-1) + (4)(5) = -2 + 20 = 18 \] This satisfies the condition for the coefficient of \(xy\). ### Step 4: Write the Factored Form Using the identified values: \[ (2x + 5y)(4x - y) = 8x^{2} + 18xy - 5y^{2} \] ### **Final Answer** \[ 8x^{2} + 18xy - 5y^{2} = (2x + 5y)(4x - y) \]