To factor the quadratic expression 9x² – 3x – 2, we first look for two numbers that multiply to the product of the leading coefficient (9) and the constant term (-2), which is -18, and also add up to the middle coefficient (-3).
The numbers that satisfy these conditions are -6 and 3, since (-6) * (3) = -18 and (-6) + (3) = -3.
Next, we can rewrite the expression by splitting the middle term:
9x² – 6x + 3x – 2.
Now, we can factor by grouping:
From the first two terms (9x² – 6x), we can factor out 3x:
3x(3x – 2).
From the last two terms (3x – 2), we can factor out 1:
1(3x – 2).
Now we have:
3x(3x – 2) + 1(3x – 2).
Both terms contain the common factor (3x – 2), so we can factor that out:
(3x – 2)(3x + 1).
Thus, the factors of the expression 9x² – 3x – 2 are (3x – 2) and (3x + 1).