\( \operatorname { ar } : ( 2 x + 3 y - 4 ) ( - 3 x + y + 2 ) \)

To find the expression for \( \operatorname{ar} = (2x + 3y - 4)(-3x + y + 2) \), you can use the distributive property (also known as the FOIL method for binomials) to expand the expression. Start by multiplying each term in the first polynomial by each term in the second polynomial: \[ = 2x \cdot (-3x) + 2x \cdot y + 2x \cdot 2 + 3y \cdot (-3x) + 3y \cdot y + 3y \cdot 2 - 4 \cdot (-3x) - 4 \cdot y - 4 \cdot 2 \] Now, calculate each product: \[ = -6x^2 + 2xy + 4x - 9xy + 3y^2 + 6 - 12x - 4y - 8 \] Next, combine like terms: \[ = -6x^2 + (2xy - 9xy) + (4x - 12x) + 3y^2 + (-4y) + (6 - 8) \] This simplifies to: \[ = -6x^2 - 7xy - 8x + 3y^2 - 2 \] So the final expanded form of \( \operatorname{ar} \) is: \[ -6x^2 - 7xy - 8x + 3y^2 - 2 \]