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

Divide the polynomials by following steps: - step0: Using Long Division to Divide Polynomials: \(\frac{\left(3x^{2}+18x+24\right)}{\left(2x-1\right)}\) - step1: Calculate: \(\frac{3}{2}x+\frac{39}{4}+\frac{\frac{135}{4}}{2x-1}\) Calculate or simplify the expression \( (x^2 + x - 12) / (4*x - 2) / ((3*x^2 + 18*x + 24) / (2*x - 1)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(x^{2}+x-12\right)}{\left(4x-2\right)}}{\left(\frac{\left(3x^{2}+18x+24\right)}{\left(2x-1\right)}\right)}\) - step1: Remove the parentheses: \(\frac{\frac{x^{2}+x-12}{4x-2}}{\frac{3x^{2}+18x+24}{2x-1}}\) - step2: Multiply by the reciprocal: \(\frac{x^{2}+x-12}{4x-2}\times \frac{2x-1}{3x^{2}+18x+24}\) - step3: Rewrite the expression: \(\frac{\left(x+4\right)\left(x-3\right)}{4x-2}\times \frac{2x-1}{\left(x+4\right)\left(3x+6\right)}\) - step4: Reduce the fraction: \(\frac{x-3}{4x-2}\times \frac{2x-1}{3x+6}\) - step5: Rewrite the expression: \(\frac{x-3}{2\left(2x-1\right)}\times \frac{2x-1}{3x+6}\) - step6: Reduce the fraction: \(\frac{x-3}{2}\times \frac{1}{3x+6}\) - step7: Multiply the terms: \(\frac{x-3}{2\left(3x+6\right)}\) - step8: Simplify: \(\frac{x-3}{6x+12}\) To simplify the expression \[ \frac{x^{2}+x-12}{4 x-2} \div \frac{3 x^{2}+18 x+24}{2 x-1}, \] we can follow these steps: 1. **Rewrite the division as multiplication by the reciprocal**: \[ \frac{x^{2}+x-12}{4 x-2} \times \frac{2 x-1}{3 x^{2}+18 x+24}. \] 2. **Simplify each part**: - The first part is \(\frac{x^{2}+x-12}{4 x-2}\). - The second part is \(\frac{2 x-1}{3 x^{2}+18 x+24}\). 3. **Simplify \(\frac{x^{2}+x-12}{4 x-2}\)**: - The numerator \(x^{2}+x-12\) factors to \((x-3)(x+4)\). - The denominator \(4x-2\) can be factored as \(2(2x-1)\). - Thus, we have: \[ \frac{(x-3)(x+4)}{2(2x-1)}. \] 4. **Simplify \(\frac{3 x^{2}+18 x+24}{2 x-1}\)**: - The numerator \(3x^{2}+18x+24\) can be factored as \(3(x^{2}+6x+8) = 3(x+2)(x+4)\). - The denominator remains \(2x-1\). - Thus, we have: \[ \frac{3(x+2)(x+4)}{2x-1}. \] 5. **Combine the simplified parts**: Now substituting back into our expression: \[ \frac{(x-3)(x+4)}{2(2x-1)} \times \frac{2x-1}{3(x+2)(x+4)}. \] 6. **Cancel common factors**: - The \(2x-1\) cancels out. - The \(x+4\) cancels out. 7. **Final simplification**: We are left with: \[ \frac{x-3}{6(x+2)}. \] Thus, the simplified form of the original expression is: \[ \frac{x-3}{6(x+2)}. \]