factorize the axpression \[ 2 y^{2}+x y-3 x^{2} \]

To factor the expression \( 2y^{2} + xy - 3x^{2} \), we're looking for two binomials that multiply to give this quadratic. First, we can rearrange the expression into a standard form which hints at the factorization technique, \( ay^2 + by + c \). The factorization process involves finding two numbers that multiply to \( ac \) (where \( a = 2 \), \( b = 1 \), and \( c = -3 \), so \( ac = -6 \)) and add to \( b \) (which is 1). The numbers are 3 and -2. We can now rewrite the middle term using these numbers: \[ 2y^{2} + 3xy - 2xy - 3x^{2} \] Next, group the terms: \[ (2y^{2} + 3xy) + (-2xy - 3x^{2}) \] This factors to: \[ y(2y + 3x) - x(2y + 3x) \] Now factor out \( (2y + 3x) \): \[ (2y + 3x)(y - x) \] So, the fully factorized form of the expression \( 2y^{2} + xy - 3x^{2} \) is: \[ (2y + 3x)(y - x) \]