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Which line shows the correct and complete distribution for the problem below?

(2x+3)(3x−1)
(1 point)
Responses

2x⋅3x +3⋅3x
2x⋅3x +3⋅3x

2x⋅3+3x⋅−1
2 x times 3 plus 3 x times negative 1

2x⋅3x+2x⋅−1+3⋅3x+3⋅−1
2 x times 3 x plus 2 x times negative 1 plus 3 times 3 x plus 3 times negative 1

2x⋅3x−2x⋅−1−3⋅3x−3⋅−1

1 answer

To correctly distribute the expression \((2x + 3)(3x - 1)\), we need to apply 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:

  1. Multiply \(2x\) by \(3x\)
  2. Multiply \(2x\) by \(-1\)
  3. Multiply \(3\) by \(3x\)
  4. Multiply \(3\) by \(-1\)

This gives us:

  • \(2x \cdot 3x\)
  • \(2x \cdot -1\)
  • \(3 \cdot 3x\)
  • \(3 \cdot -1\)

Now, putting this all together, we have:

\[ 2x \cdot 3x + 2x \cdot -1 + 3 \cdot 3x + 3 \cdot -1 \]

Therefore, the correct and complete distribution, based on the responses provided, is:

2x⋅3x + 2x⋅−1 + 3⋅3x + 3⋅−1

So the choice that represents the correct and complete distribution for the original problem is:

2x⋅3x + 2x⋅−1 + 3⋅3x + 3⋅−1