\( [ ( 3 x ^ { 2 } + 5 x y ) - ( 6 x ^ { 2 } - 4 x y ) ] \div 4 x . \)

Let's simplify the given expression step by step: \[ \frac{(3x^{2} + 5xy) - (6x^{2} - 4xy)}{4x} \] ### Step 1: Expand the Numerator First, distribute the negative sign to the terms inside the second parentheses: \[ 3x^{2} + 5xy - 6x^{2} + 4xy \] Combine like terms: \[ (3x^{2} - 6x^{2}) + (5xy + 4xy) = -3x^{2} + 9xy \] So, the expression now is: \[ \frac{-3x^{2} + 9xy}{4x} \] ### Step 2: Factor the Numerator Factor out a common factor from the numerator: \[ -3x^{2} + 9xy = 3(-x^{2} + 3xy) = 3x(-x + 3y) \] Now, the expression becomes: \[ \frac{3x(-x + 3y)}{4x} \] ### Step 3: Simplify the Expression Cancel out the common factor of \( x \) in the numerator and the denominator: \[ \frac{3 \cancel{x}(-x + 3y)}{4 \cancel{x}} = \frac{3(-x + 3y)}{4} \] You can also write it as: \[ \frac{9y - 3x}{4} \] ### Final Simplified Form \[ \frac{9y - 3x}{4} \quad \text{or} \quad \frac{3(3y - x)}{4} \]