Factor \( 6 x^{2}-19 x-20 \) \( (2 x-5)(3 x+4) \) \( (6 x+5)(x-4) \) \( (6 x-4)(x+5) \) \( (2 x+5)(3 x-4) \)

To factor the quadratic expression \( 6x^2 - 19x - 20 \), we can begin by applying the method of finding two numbers that multiply to \( ac \) (where \( a = 6 \) and \( c = -20 \)) and add up to \( b = -19 \). Here, \( ac = 6 \times -20 = -120 \). We seek two numbers that multiply to -120 and add up to -19. The numbers are -24 and 5. We can rewrite the middle term and factor by grouping. Rearranging gives us: \( 6x^2 - 24x + 5x - 20 \). Now, grouping the terms yields: \( (6x^2 - 24x) + (5x - 20) \), which factors to \( 6x(x - 4) + 5(x - 4) = (6x + 5)(x - 4) \). This means the correct factorization is \( (6x + 5)(x - 4) \). A fun little fact about factoring quadratics is that it’s like solving a mystery! Each number plays a key role, and by piecing clues (the coefficients) together, you discover the hidden relationship—like finding a secret passage in a storybook! If you're delving deeper into factoring quadratics, consider exploring how the discriminant from the quadratic formula gives insight into the nature of the roots. A positive discriminant means two distinct real roots, zero means one real root, and a negative discriminant indicates complex roots. It's an essential concept that can transform your understanding of quadratic functions!