To correctly expand the expression (2x + 3)(3x - 1), we will use the distributive property (also known as the FOIL method for binomials). This means we will multiply each term in the first binomial by each term in the second binomial.
Let's break it down:
- Multiply the first terms: \(2x \cdot 3x = 6x^2\)
- Multiply the outer terms: \(2x \cdot -1 = -2x\)
- Multiply the inner terms: \(3 \cdot 3x = 9x\)
- Multiply the last terms: \(3 \cdot -1 = -3\)
Now, combining these results:
\[ 6x^2 - 2x + 9x - 3 \]
Combine the like terms:
\[ 6x^2 + 7x - 3 \]
Now to determine which option corresponds to the correct distribution:
- Option A: \(2x \cdot 3x + 3 \cdot 3x\) (Incorrect, does not cover all terms)
- Option B: \(2x \cdot 3 + 3x \cdot -1\) (Incorrect, does not cover all terms)
- Option C: \(2x \cdot 3x + 2x \cdot -1 + 3 \cdot 3x + 3 \cdot -1\) (Correct, this represents the complete distribution)
- Option D: \(2x \cdot 3x - 2x \cdot -1 - 3 \cdot 3x - 3 \cdot -1\) (Incorrect, includes incorrect signs and terms)
The correct answer is C: \(2x \cdot 3x + 2x \cdot -1 + 3 \cdot 3x + 3 \cdot -1\).