mplify \( \frac{\left(3 x^{2}-13 x+10\right)}{\left(2 x^{2}-16 x+30\right)} \div \frac{\left(3 x^{2}+5 x+2\right)}{\left(x^{3}-2 x^{2}-3 x\right)} \)

To simplify the expression: \[ \frac{\left(3x^{2} - 13x + 10\right)}{\left(2x^{2} - 16x + 30\right)} \div \frac{\left(3x^{2} + 5x + 2\right)}{\left(x^{3} - 2x^{2} - 3x\right)} \] follow these steps: ### Step 1: Rewrite the Division as Multiplication Dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, the expression becomes: \[ \frac{3x^{2} - 13x + 10}{2x^{2} - 16x + 30} \times \frac{x^{3} - 2x^{2} - 3x}{3x^{2} + 5x + 2} \] ### Step 2: Factor Each Polynomial **1. Factor \(3x^{2} - 13x + 10\):** Find two numbers that multiply to \(3 \times 10 = 30\) and add up to \(-13\). These numbers are \(-10\) and \(-3\). \[ 3x^{2} - 13x + 10 = 3x^{2} - 10x - 3x + 10 = (x - 1)(3x - 10) \] **2. Factor \(2x^{2} - 16x + 30\):** Factor out the greatest common factor (2): \[ 2x^{2} - 16x + 30 = 2(x^{2} - 8x + 15) \] Now, factor \(x^{2} - 8x + 15\): \[ x^{2} - 8x + 15 = (x - 5)(x - 3) \] So, \[ 2x^{2} - 16x + 30 = 2(x - 5)(x - 3) \] **3. Factor \(3x^{2} + 5x + 2\):** Find two numbers that multiply to \(3 \times 2 = 6\) and add up to \(5\). These numbers are \(2\) and \(3\). \[ 3x^{2} + 5x + 2 = 3x^{2} + 3x + 2x + 2 = (3x + 2)(x + 1) \] **4. Factor \(x^{3} - 2x^{2} - 3x\):** Factor out the greatest common factor (x): \[ x^{3} - 2x^{2} - 3x = x(x^{2} - 2x - 3) \] Now, factor \(x^{2} - 2x - 3\): \[ x^{2} - 2x - 3 = (x - 3)(x + 1) \] So, \[ x^{3} - 2x^{2} - 3x = x(x - 3)(x + 1) \] ### Step 3: Substitute the Factored Forms Back Into the Expression \[ \frac{(x - 1)(3x - 10)}{2(x - 5)(x - 3)} \times \frac{x(x - 3)(x + 1)}{(3x + 2)(x + 1)} \] ### Step 4: Cancel Common Factors - **Cancel \((x - 3)\):** Appears in both the numerator and the denominator. - **Cancel \((x + 1)\):** Appears in both the numerator and the denominator. After canceling, the expression simplifies to: \[ \frac{(x - 1)(3x - 10) \cdot x}{2(x - 5)(3x + 2)} \] ### Final Simplified Expression \[ \boxed{\frac{x(x - 1)(3x - 10)}{2(x - 5)(3x + 2)}} \]