To factor the quadratic expression 3x^2 - x - 2, we need to find two binomials that, when multiplied, give us the original expression.
The factored form of the quadratic expression can be determined by breaking down the middle term (-x) into two terms whose coefficients multiply to give the product of the coefficient of the squared term (3) and the constant term (-2). In this case, the product is -6. We are looking for two numbers whose sum is equal to -1 (the coefficient of the middle term) and whose product is equal to -6.
The numbers that satisfy these conditions are -3 and 2. We can now rewrite the expression using these numbers:
3x^2 - x - 2 = 3x^2 - 3x + 2x - 2
Next, we group the terms and factor by grouping:
(3x^2 - 3x) + (2x - 2) = 3x(x - 1) + 2(x - 1)
Now, we can see that we have a common binomial factor of (x - 1) in both terms. We can factor this out:
3x(x - 1) + 2(x - 1) = (3x + 2)(x - 1)
Therefore, the factored form of the quadratic expression 3x^2 - x - 2 is (3x + 2)(x - 1).