Problem 15: \[ \frac{3 x-6}{2 x^{2}} \div \frac{x^{2}-5 x+6}{4 x-12} \]

To solve the expression \[ \frac{3 x-6}{2 x^{2}} \div \frac{x^{2}-5 x+6}{4 x-12}, \] first, we'll rewrite the division as multiplication by the reciprocal: \[ \frac{3 x-6}{2 x^{2}} \times \frac{4 x - 12}{x^{2} - 5 x + 6}. \] Next, let's factor the components: 1. In the numerator of the first fraction: \[ 3x - 6 = 3(x - 2). \] 2. In the numerator of the second fraction: \[ 4x - 12 = 4(x - 3). \] 3. In the denominator of the second fraction, we can factor \(x^2 - 5x + 6\): \[ x^{2}-5x+6 = (x - 2)(x - 3). \] Now, substituting these factorizations back into the expression, we get: \[ \frac{3(x-2)}{2x^2} \times \frac{4(x - 3)}{(x - 2)(x - 3)}. \] We can now cancel out common factors: - \(x - 2\) from the numerator of the first fraction with the denominator of the second fraction. - \(x - 3\) from the numerator of the second fraction with the denominator of the second fraction. Thus, the expression simplifies to: \[ \frac{3 \cdot 4}{2x^2} = \frac{12}{2x^2} = \frac{6}{x^2}. \] So, the final simplified result is: \[ \frac{6}{x^2}. \]