To find the correct and complete distribution of the expression \((2x + 3)(3x - 1)\), we need to apply the distributive property, also known as the FOIL method for binomials.
The expression can be distributed as follows:
- First term: \(2x \cdot 3x = 6x^2\)
- Outer term: \(2x \cdot -1 = -2x\)
- Inner term: \(3 \cdot 3x = 9x\)
- Last term: \(3 \cdot -1 = -3\)
Combining these results gives: \[ 6x^2 - 2x + 9x - 3 \] which simplifies to: \[ 6x^2 + 7x - 3 \]
Now, looking at the options provided:
A) \(2x \cdot 3x + 3 \cdot 3x\)
B) \(2x \cdot 3x + 3x \cdot -1\)
C) \(2x \cdot 3x + 2x \cdot -1 + 3 \cdot 3x + 3 \cdot -1\)
D) \(2x \cdot 3x - 2x \cdot -1 - 3 \cdot 3x \cdot -1\)
The correct and complete distribution should include all four products we calculated.
Therefore, the correct answer is C, which includes:
- \(2x \cdot 3x\) (first term)
- \(2x \cdot -1\) (outer term)
- \(3 \cdot 3x\) (inner term)
- \(3 \cdot -1\) (last term)
Thus, option C shows the correct and complete distribution.
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