To distribute the expression \((2x + 3)(3x - 1)\), we use the distributive property (also known as the FOIL method for binomials). This means we multiply each term in the first parentheses by each term in the second parentheses.
The steps are as follows:
- Multiply \(2x\) by \(3x\).
- Multiply \(2x\) by \(-1\).
- Multiply \(3\) by \(3x\).
- Multiply \(3\) by \(-1\).
Now, let's calculate:
- \(2x \cdot 3x = 6x^2\)
- \(2x \cdot (-1) = -2x\)
- \(3 \cdot 3x = 9x\)
- \(3 \cdot (-1) = -3\)
Now, combine all these results:
\[ 6x^2 + (-2x) + 9x + (-3) = 6x^2 + 7x - 3 \]
Now, let's analyze the lines presented:
- Line 1: \(2x \cdot 3x + 3 \cdot 3x\) (not correct, misses other products)
- Line 2: \(2x \cdot 3 + 3x \cdot (-1)\) (not correct, misses products)
- Line 3: \(2x \cdot 3x + 2x \cdot (-1) + 3 \cdot 3x + 3 \cdot (-1)\) (this is the correct and complete distribution)
- Line 4: \(2x \cdot 3x - 2x \cdot (-1) - 3 \cdot 3x - 3 \cdot (-1)\) (incorrect signs)
Based on this analysis, the correct line showing the correct and complete distribution is Line 3:
\[ 2x \cdot 3x + 2x \cdot (-1) + 3 \cdot 3x + 3 \cdot (-1) \]
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