Final answer:
To factor the expression 2x^2 - x - 3, one must find two numbers that multiply to -6 and add up to -1. The numbers are -3 and +2, making the factored form (2x - 3)(x + 1). One can verify the factoring by expanding the binomials to show they equal the original expression.
Step-by-step explanation:
To factor the expression 2x^2 - x - 3, you need to find two numbers that both multiply to give the product of the quadratic term coefficient (2) and the constant term (-3), which is -6, and add up to give the linear term coefficient (-1). The process of factoring a quadratic expression involves finding two binomials that when multiplied together will give us the original quadratic expression.
In this case, we are looking for two numbers that multiply to -6 and add up to -1. The numbers that meet these criteria are -3 and +2. Therefore, the factored form of 2x^2 - x - 3 is (2x - 3)(x + 1).
We can verify this by expanding the binomials:
- 2x * x = 2x^2
- 2x * 1 = 2x
- -3 * x = -3x
- -3 * 1 = -3
Combining like terms (2x - 3x) gives us -x, so the expanded form is 2x^2 - x - 3, confirming that (2x - 3)(x + 1) is indeed the factored form.