To factor the quadratic expression \(3x^2 - x - 2\), we are looking for two numbers that multiply to \(3 \times -2 = -6\) and add to \(-1\) (the coefficient of \(x\)).
The two numbers that satisfy these conditions are \(2\) and \(-3\).
We can rewrite the middle term \(-x\) using these two numbers:
\[ 3x^2 + 2x - 3x - 2 \]
Next, we can group the terms:
\[ (3x^2 + 2x) + (-3x - 2) \]
Factoring out the common factors in each group gives:
\[ x(3x + 2) - 1(3x + 2) \]
Now, we can factor out the common binomial \(3x + 2\):
\[ (3x + 2)(x - 1) \]
So, the factorization of the quadratic expression \(3x^2 - x - 2\) is:
\[ (3x + 2)(x - 1) \]