When factoring the quadratic expression 6x^2 + x - 15, what is the correct factorization?

To factor the quadratic expression 6x² + x - 15, we are looking for two binomials that will multiply back to yield the original quadratic. The general strategy in factoring quadratics is to find two numbers that multiply to give the product of the leading coefficient and the constant term, while also adding up to the middle coefficient.

First, we calculate the product of the leading coefficient (6) and the constant term (-15), which gives us -90. Then, we look for two numbers that multiply to -90 and add to the middle coefficient, which is +1 (from the x term). The appropriate pair of numbers that meet these criteria is 10 and -9 since 10 * -9 = -90 and 10 + (-9) = 1.

Using these numbers, we can rewrite the middle term (x) as a sum of two terms: 6x² + 10x - 9x - 15. Next, we can group the terms to factor by grouping:

1. Group the first two terms and the last two terms: (6x² + 10x) + (-9x - 15).

2. Factor out the common factors: 2x(3