\( 6 x ^ { 2 } + 5 x y - 15 y ^ { 2 } \)

The expression you've provided is a quadratic in two variables: \[ 6x^2 + 5xy - 15y^2 \] ### **Attempting to Factor the Expression** To factor a quadratic trinomial of the form \( ax^2 + bxy + cy^2 \), we look for two binomials: \[ (ax + by)(cx + dy) \] Expanding these binomials gives: \[ acx^2 + (ad + bc)xy + bdy^2 \] For your expression: - \( ac = 6 \) - \( bd = -15 \) - \( ad + bc = 5 \) **Possible Factor Pairs:** 1. **For \( ac = 6 \):** - (2, 3) - (3, 2) - (6, 1) - (1, 6) 2. **For \( bd = -15 \):** - (1, -15) - (-1, 15) - (3, -5) - (-3, 5) - (5, -3) - (-5, 3) **Testing Factor Combinations:** After systematically testing all possible combinations of these factors, none satisfy the condition \( ad + bc = 5 \). For example: - **Using \( a = 2 \) and \( c = 3 \):** - If \( b = 5 \) and \( d = -3 \): \[ ad + bc = 2(-3) + 3(5) = -6 + 15 = 9 \neq 5 \] - Similarly, other combinations do not yield \( ad + bc = 5 \). **Conclusion:** Since none of the integer or simple fractional combinations satisfy all the necessary conditions for factoring, the quadratic expression: \[ 6x^2 + 5xy - 15y^2 \] **cannot be factored into binomials with rational coefficients**. ### **Alternative Approach: Completing the Square** If you're interested in other forms, such as completing the square or analyzing the expression further, please let me know!