3. Simplify \[ \frac{x^{2}+x-12}{4 x-2} \div \frac{3 x^{2}+18 x+24}{2 x-1} \]

To simplify the expression \[ \frac{x^{2}+x-12}{4 x-2} \div \frac{3 x^{2}+18 x+24}{2 x-1}, \] we start by rewriting the division as multiplication by the reciprocal. Therefore, we have: \[ \frac{x^{2}+x-12}{4 x-2} \times \frac{2 x-1}{3 x^{2}+18 x+24}. \] Next, let’s factor each polynomial: 1. **Factor \(x^{2} + x - 12\)**: - This can be factored into \((x + 4)(x - 3)\). 2. **Factor \(4x - 2\)**: - This can be factored out as \(2(2x - 1)\). 3. **Factor \(3x^{2} + 18x + 24\)**: - We can factor out a common factor of \(3\), giving us \(3(x^{2} + 6x + 8)\). - The quadratic \(x^{2} + 6x + 8\) can be factored into \((x + 2)(x + 4)\). 4. **Factor \(2x - 1\)**: - This is already in factored form. Putting this all into our expression, we have: \[ \frac{(x + 4)(x - 3)}{2(2x - 1)} \times \frac{2x - 1}{3(x + 2)(x + 4)}. \] Now we can cancel common terms. The \(2x - 1\) cancels with itself, and \(x + 4\) also cancels: \[ \frac{(x - 3)}{2 \cdot 3(x + 2)} = \frac{x - 3}{6(x + 2)}. \] Thus, the simplified expression is: \[ \frac{x - 3}{6(x + 2)}. \]