To factor the quadratic expression \(6x^2 + 21x - 12\), we can start by looking for two numbers that multiply to \(6 \cdot (-12) = -72\) and add to \(21\) (the coefficient of \(x\)).
We find that the numbers \(24\) and \(-3\) fit these criteria, since:
- \(24 \times (-3) = -72\)
- \(24 + (-3) = 21\)
Next, we can rewrite the middle term \(21x\) using these numbers:
\[ 6x^2 + 24x - 3x - 12 \]
Now, we can group the terms:
\[ (6x^2 + 24x) + (-3x - 12) \]
Factoring out the common factors in each group gives us:
\[ 6x(x + 4) - 3(x + 4) \]
Now, we can factor out the common binomial factor \((x + 4)\):
\[ (6x - 3)(x + 4) \]
We can also factor \(6x - 3\) further:
\[ 3(2x - 1)(x + 4) \]
Thus, the factors of \(6x^2 + 21x - 12\) are:
- \(x + 4\)
- \(2x - 1\)
- \(3\)
Any combination of the three factors listed above can potentially work as a correct answer depending on the options given.
The correct factors of the expression are \(3\), \(x + 4\), and \(2x - 1\).
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