To factor the quadratic expression \(9x^2 + 3x - 2\), we can use the method of factoring by grouping or apply the quadratic formula if necessary. First, we look for two numbers that multiply to \(ac\) (where \(a\) is the coefficient of \(x^2\) and \(c\) is the constant term) and add to \(b\) (coefficient of \(x\)).
For \(9x^2 + 3x - 2\):
- \(a = 9\)
- \(b = 3\)
- \(c = -2\)
We calculate \(ac = 9 \cdot (-2) = -18\).
Next, we need to find two numbers that multiply to \(-18\) and add to \(3\). Those numbers are \(6\) and \(-3\).
Now we rewrite the middle term (\(3x\)) using \(6x\) and \(-3x\):
\[ 9x^2 + 6x - 3x - 2 \]
Next, we group the terms:
\[ (9x^2 + 6x) + (-3x - 2) \]
Now we factor each group:
\[ 3x(3x + 2) - 1(3x + 2) \]
Now, we can factor out the common factor \((3x + 2)\):
\[ (3x + 2)(3x - 1) \]
Thus, the expression \(9x^2 + 3x - 2\) can be factored as:
\[ (3x + 2)(3x - 1) \]
This is the final factored form of the expression.