Divide and simplify. \[ \frac{2 x^{2}+15 x-38}{4 x^{2}+11 x-3} \div \frac{3 x^{2}-4 x-4}{3 x^{2}+11 x+6} \]

Factor the expression by following steps: - step0: Factor: \(4x^{2}+11x-3\) - step1: Rewrite the expression: \(4x^{2}+\left(-1+12\right)x-3\) - step2: Calculate: \(4x^{2}-x+12x-3\) - step3: Rewrite the expression: \(x\times 4x-x+3\times 4x-3\) - step4: Factor the expression: \(x\left(4x-1\right)+3\left(4x-1\right)\) - step5: Factor the expression: \(\left(x+3\right)\left(4x-1\right)\) Factor the expression \( 2 x^{2}+15 x-38 \). Factor the expression by following steps: - step0: Factor: \(2x^{2}+15x-38\) - step1: Rewrite the expression: \(2x^{2}+\left(19-4\right)x-38\) - step2: Calculate: \(2x^{2}+19x-4x-38\) - step3: Rewrite the expression: \(x\times 2x+x\times 19-2\times 2x-2\times 19\) - step4: Factor the expression: \(x\left(2x+19\right)-2\left(2x+19\right)\) - step5: Factor the expression: \(\left(x-2\right)\left(2x+19\right)\) Factor the expression \( 3 x^{2}+11 x+6 \). Factor the expression by following steps: - step0: Factor: \(3x^{2}+11x+6\) - step1: Rewrite the expression: \(3x^{2}+\left(2+9\right)x+6\) - step2: Calculate: \(3x^{2}+2x+9x+6\) - step3: Rewrite the expression: \(x\times 3x+x\times 2+3\times 3x+3\times 2\) - step4: Factor the expression: \(x\left(3x+2\right)+3\left(3x+2\right)\) - step5: Factor the expression: \(\left(x+3\right)\left(3x+2\right)\) Factor the expression \( 3 x^{2}-4 x-4 \). Factor the expression by following steps: - step0: Factor: \(3x^{2}-4x-4\) - step1: Rewrite the expression: \(3x^{2}+\left(2-6\right)x-4\) - step2: Calculate: \(3x^{2}+2x-6x-4\) - step3: Rewrite the expression: \(x\times 3x+x\times 2-2\times 3x-2\times 2\) - step4: Factor the expression: \(x\left(3x+2\right)-2\left(3x+2\right)\) - step5: Factor the expression: \(\left(x-2\right)\left(3x+2\right)\) Calculate or simplify the expression \( \frac{(x-2)(2x+19)}{(x+3)(4x-1)} \div \frac{(x-2)(3x+2)}{(x+3)(3x+2)} \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x-2\right)\left(2x+19\right)}{\left(x+3\right)\left(4x-1\right)}\div \frac{\left(x-2\right)\left(3x+2\right)}{\left(x+3\right)\left(3x+2\right)}\) - step1: Reduce the fraction: \(\frac{\left(x-2\right)\left(2x+19\right)}{\left(x+3\right)\left(4x-1\right)}\div \frac{x-2}{x+3}\) - step2: Multiply by the reciprocal: \(\frac{\left(x-2\right)\left(2x+19\right)}{\left(x+3\right)\left(4x-1\right)}\times \frac{x+3}{x-2}\) - step3: Reduce the fraction: \(\frac{2x+19}{4x-1}\times 1\) - step4: Multiply the terms: \(\frac{2x+19}{4x-1}\) To simplify the expression \[ \frac{2 x^{2}+15 x-38}{4 x^{2}+11 x-3} \div \frac{3 x^{2}-4 x-4}{3 x^{2}+11 x+6}, \] we first factor each polynomial involved. 1. **Factoring the first numerator**: \[ 2x^2 + 15x - 38 = (x - 2)(2x + 19). \] 2. **Factoring the first denominator**: \[ 4x^2 + 11x - 3 = (x + 3)(4x - 1). \] 3. **Factoring the second numerator**: \[ 3x^2 - 4x - 4 = (x - 2)(3x + 2). \] 4. **Factoring the second denominator**: \[ 3x^2 + 11x + 6 = (x + 3)(3x + 2). \] Now, we can rewrite the original expression using these factors: \[ \frac{(x-2)(2x+19)}{(x+3)(4x-1)} \div \frac{(x-2)(3x+2)}{(x+3)(3x+2)}. \] Dividing by a fraction is equivalent to multiplying by its reciprocal: \[ \frac{(x-2)(2x+19)}{(x+3)(4x-1)} \cdot \frac{(x+3)(3x+2)}{(x-2)(3x+2)}. \] Next, we can cancel out the common factors: - The factor \( (x - 2) \) cancels from the numerator and denominator. - The factor \( (x + 3) \) cancels from the numerator and denominator. This simplifies our expression to: \[ \frac{2x + 19}{4x - 1}. \] Thus, the final simplified result is: \[ \frac{2x + 19}{4x - 1}. \]