Perform the indicated operation. \[ \frac{2 x^{2}-6 x-8}{3 x^{2}+6 x-6} \div \frac{x^{2}-5 x+4}{x^{2}+4 x-5} \]

Factor the expression by following steps: - step0: Factor: \(3x^{2}+6x-6\) - step1: Factor the expression: \(3\left(x^{2}+2x-2\right)\) Factor the expression \( x^{2}+4 x-5 \). Factor the expression by following steps: - step0: Factor: \(x^{2}+4x-5\) - step1: Rewrite the expression: \(x^{2}+\left(5-1\right)x-5\) - step2: Calculate: \(x^{2}+5x-x-5\) - step3: Rewrite the expression: \(x\times x+x\times 5-x-5\) - step4: Factor the expression: \(x\left(x+5\right)-\left(x+5\right)\) - step5: Factor the expression: \(\left(x-1\right)\left(x+5\right)\) Factor the expression \( x^{2}-5 x+4 \). Factor the expression by following steps: - step0: Factor: \(x^{2}-5x+4\) - step1: Rewrite the expression: \(x^{2}+\left(-1-4\right)x+4\) - step2: Calculate: \(x^{2}-x-4x+4\) - step3: Rewrite the expression: \(x\times x-x-4x+4\) - step4: Factor the expression: \(x\left(x-1\right)-4\left(x-1\right)\) - step5: Factor the expression: \(\left(x-4\right)\left(x-1\right)\) Factor the expression \( 2 x^{2}-6 x-8 \). Factor the expression by following steps: - step0: Factor: \(2x^{2}-6x-8\) - step1: Rewrite the expression: \(2x^{2}-2\times 3x-2\times 4\) - step2: Factor the expression: \(2\left(x^{2}-3x-4\right)\) - step3: Factor the expression: \(2\left(x-4\right)\left(x+1\right)\) Let's start by factoring each expression in the given operation: 1. **Numerator of the first fraction**: \[ 2x^2 - 6x - 8 = 2(x - 4)(x + 1) \] 2. **Denominator of the first fraction**: \[ 3x^2 + 6x - 6 = 3(x^2 + 2x - 2) \] 3. **Numerator of the second fraction**: \[ x^2 - 5x + 4 = (x - 4)(x - 1) \] 4. **Denominator of the second fraction**: \[ x^2 + 4x - 5 = (x - 1)(x + 5) \] Now, we can rewrite the original expression using these factored forms: \[ \frac{2(x - 4)(x + 1)}{3(x^2 + 2x - 2)} \div \frac{(x - 4)(x - 1)}{(x - 1)(x + 5)} \] To perform the division, we multiply by the reciprocal of the second fraction: \[ \frac{2(x - 4)(x + 1)}{3(x^2 + 2x - 2)} \times \frac{(x - 1)(x + 5)}{(x - 4)(x - 1)} \] Now, we can simplify the expression: - The \((x - 4)\) terms cancel out. - The \((x - 1)\) terms also cancel out. This leaves us with: \[ \frac{2(x + 1)(x + 5)}{3(x^2 + 2x - 2)} \] Next, we need to simplify the denominator \(3(x^2 + 2x - 2)\). Let's factor \(x^2 + 2x - 2\): \[ x^2 + 2x - 2 = (x + 1)^2 - 3 = (x + 1 - \sqrt{3})(x + 1 + \sqrt{3}) \] Thus, the expression becomes: \[ \frac{2(x + 1)(x + 5)}{3((x + 1 - \sqrt{3})(x + 1 + \sqrt{3}))} \] Now, we can simplify further if needed, but this is the final simplified form of the expression: \[ \frac{2(x + 5)}{3((x + 1 - \sqrt{3})(x + 1 + \sqrt{3}))} \] This is the result of the operation.