Factor completely. Then select the correct factors from the options below. (You will select two factors). \( 4 x^{2}-5 x-6 \)

To factor the quadratic \( 4x^2 - 5x - 6 \) completely, we first look for two numbers that multiply to \( 4 \times -6 = -24 \) and add up to \( -5 \). These numbers are \( -8 \) and \( 3 \). We can rewrite the equation as \( 4x^2 - 8x + 3x - 6 \), then group it as \( 4x(x - 2) + 3(x - 2) \), leading us to factor it as \( (4x + 3)(x - 2) \). Now, if you're picking factors, you would select \( (4x + 3) \) and \( (x - 2) \). This means that you effectively turned a puzzle into a masterpiece! Quadratics like this can often feel daunting, but once you break them down into manageable pieces, they become a lot easier to digest. Factorization is like unraveling a mystery — every equation holds its secrets waiting to be revealed! If you love diving into more advanced realms of algebra, consider tackling polynomials of higher degree or even exploring complex numbers. You'll soon find that the world of mathematics is filled with opportunities for discovery, and every new concept you master opens up a treasure trove of future challenges!