What are the factors of 2x^2 + xy - 3y^2?

To determine the correct factors of the expression 2x^2 + xy - 3y^2, we should first consider how to factor quadratic expressions generally.

We want to express the quadratic in the form (ax + by)(cx + dy) where the product leads us back to the original expression. In this case, we look for two binomials that will yield the first term of 2x^2, the middle term xy, and the last term of -3y^2 when multiplied out.

Taking a closer look at the proposed correct answer, (2x + 3y)(x - y), we can verify it through distribution:

1. First, distribute 2x across the second binomial:

   • 2x * x = 2x^2
   • 2x * -y = -2xy

2. Next, distribute 3y across the second binomial:

   • 3y * x = 3xy
   • 3y * -y = -3y^2

3. Combine all these terms:

   • 2x^2 - 2xy + 3xy - 3y^2 = 2x^