To find the correct distribution of the expression \((2x+3)(3x-1)\), we apply the distributive property (also known as the FOIL method for binomials), which includes multiplying each term in the first binomial by each term in the second binomial:
- Multiply \(2x\) by \(3x\)
- Multiply \(2x\) by \(-1\)
- Multiply \(3\) by \(3x\)
- Multiply \(3\) by \(-1\)
Now, let’s perform these multiplications:
- \(2x \cdot 3x = 6x^2\)
- \(2x \cdot -1 = -2x\)
- \(3 \cdot 3x = 9x\)
- \(3 \cdot -1 = -3\)
Now combine all these results:
\[ 6x^2 - 2x + 9x - 3 \] Combine the \(x\) terms: \[ 6x^2 + 7x - 3 \]
Now let's look through the answer choices to find the one that corresponds accurately to this process:
A.) \(2x \cdot 3 + 3x \cdot -1\) (This does not account for the full distribution).
B.) \(2x \cdot 3x - 2x \cdot -1 - 3 \cdot 3x - 3 \cdot -1\) (This has incorrect signs and additional terms, thus is incorrect).
C.) \(2x \cdot 3x + 2x \cdot -1 + 3 \cdot 3x + 3 \cdot -1\) (This is partially correct but should not split the first term's coefficient, it should stay as \(2x\)).
D.) \(2x \cdot 3x + 3 \cdot 3x\) (This does not represent both binomials as well).
Upon reviewing these choices, the closest accurate statement is choice C, despite its format. It properly lists the relevant multiplications, thereby representing the underlying process of distribution accurately.
So, the answer is C) (with the caveat that it could be better formatted).
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