The given problem involves simplifying the following algebraic fractions:
\[ \frac{x^2 + 3x - xy - 3y}{2x^2 + 2x - 4} \div \frac{3y - 3x + xy - x^2}{x^2 + x - 2} \]
Step 1: Factorize the Numerators and Denominators
First Fraction:
- Numerator: \(x^2 + 3x - xy - 3y = (x - y)(x + 3)\)
- Denominator: \(2x^2 + 2x - 4 = 2(x + 2)(x - 1)\)
Second Fraction:
- Numerator: \(3y - 3x + xy - x^2 = -(x + 3)(x - y)\)
- Denominator: \(x^2 + x - 2 = (x + 2)(x - 1)\)
Step 2: Rewrite the Division as Multiplication by the Reciprocal
\[ \frac{(x - y)(x + 3)}{2(x + 2)(x - 1)} \div \frac{-(x + 3)(x - y)}{(x + 2)(x - 1)} = \frac{(x - y)(x + 3)}{2(x + 2)(x - 1)} \times \frac{(x + 2)(x - 1)}{-(x + 3)(x - y)} \]
Step 3: Simplify the Expression
Cancel out the common factors:
\[ \frac{(x - y)(x + 3)}{2(x + 2)(x - 1)} \times \frac{(x + 2)(x - 1)}{-(x + 3)(x - y)} = \frac{1}{2} \times \frac{1}{-1} = -\frac{1}{2} \]
Final Answer:
\(-\frac{1}{2}\)