Which of the following is the correct factorization of the quadratic expression \( 12 x^{2}+14 x-10 \) ? \( (4 x+5)(3 x-2) \) \( 2(3 x+1)(2 x-5) \) \( (4 x-5)(3 x+2) \) \( 2(2 x-1)(3 x+5) \) \( 2(x+1)(6 x-5) \)

To find the correct factorization of the quadratic expression \( 12 x^{2}+14 x-10 \), let's begin by checking each option. First, we simplify the expression: 1. The expression can be factored out by 2, so we have: \[ 2(6x^2 + 7x - 5) \] 2. Now we need to factor the trinomial \( 6x^2 + 7x - 5 \). We can look for two numbers that multiply to \( 6 \times -5 = -30 \) and add up to \( 7 \). These numbers are \( 10 \) and \( -3 \). 3. Rewrite the middle term: \[ 6x^2 + 10x - 3x - 5 \] 4. Group the terms: \[ (6x^2 + 10x) + (-3x - 5) \] 5. Factor out the common factors: \[ 2x(3x + 5) - 1(3x + 5) = (3x + 5)(2x - 1) \] 6. Then multiply by \( 2 \): \[ 2(3x + 5)(2x - 1) \] So, the correct answer from the given options is \( 2(2 x-1)(3 x+5) \). Now, let's continue with more insights! The quadratic expression has roots derived from the quadratic formula, which provides a direct path to finding the x-intercepts of the quadratic. By substituting coefficients into the formula, you'll discover where this equation intersects the x-axis, leading you to insights about its graph! Another fun tip: If you're eager to improve your factoring skills, practice is key! Experiment with different quadratic expressions—try adding or changing coefficients, and see if you can factor them easily. Not only will you master the concept, but you'll also gain an appreciation for the patterns that emerge in these fascinating mathematical puzzles!