Which multiplication expression is equivalent to \( \frac{2 x^{2}-5 x-3}{4 x^{2}+12 x+5} \div \frac{3 x^{2}-11 x+6}{6 x^{2}+11 x-10} ? \) \( \frac{(x-3)(2 x+1)}{(2 x+1)(2 x+5)} \cdot \frac{(x-3)(3 x-2)}{(2 x+5)(3 x-2)} \) \( \frac{(2 x+1)(2 x+5)}{(x-3)(2 x+1)} \cdot \frac{(2 x+5)(3 x-2)}{(x-3)(3 x-2)} \) \( \frac{(x-3)(2 x+1)}{(2 x+1)(2 x+5)} \cdot \frac{(2 x+5)(3 x-2)}{(x-3)(3 x-2)} \)

We start with \[ \frac{2x^2-5x-3}{4x^2+12x+5} \div \frac{3x^2-11x+6}{6x^2+11x-10}. \] Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{2x^2-5x-3}{4x^2+12x+5} \times \frac{6x^2+11x-10}{3x^2-11x+6}. \] **Step 1. Factorize each quadratic.** 1. Factorize \(2x^2-5x-3\): - Look for two numbers whose product is \(2 \times (-3) = -6\) and whose sum is \(-5\). The numbers \(-6\) and \(1\) work. - Split the middle term: \[ 2x^2 - 6x + x - 3 = 2x(x - 3) + 1(x - 3) = (x-3)(2x+1). \] 2. Factorize \(4x^2+12x+5\): - We write: \[ 4x^2+12x+5 = (2x+1)(2x+5). \] 3. Factorize \(6x^2+11x-10\): - Look for two numbers whose product is \(6 \times (-10) = -60\) and sum is \(11\); the numbers \(15\) and \(-4\) work. - Write: \[ 6x^2+11x-10 = 6x^2+15x-4x-10 = 3x(2x+5)-2(2x+5)= (2x+5)(3x-2). \] 4. Factorize \(3x^2-11x+6\): - Look for two numbers whose product is \(3 \times 6 = 18\) and whose sum is \(-11\); the numbers \(-9\) and \(-2\) work. - Write: \[ 3x^2-11x+6 = 3x^2-9x-2x+6 = 3x(x-3)-2(x-3) = (x-3)(3x-2). \] **Step 2. Substitute the factorizations into the expression.** The expression becomes: \[ \frac{(x-3)(2x+1)}{(2x+1)(2x+5)} \times \frac{(2x+5)(3x-2)}{(x-3)(3x-2)}. \] **Step 3. Compare with the given multiplication expressions.** The expression above is exactly the third option: \[ \frac{(x-3)(2x+1)}{(2x+1)(2x+5)} \cdot \frac{(2x+5)(3x-2)}{(x-3)(3x-2)}. \] Thus, the equivalent expression is the third option.