Question

Which line shows the correct and complete distribution for the problem below? (2x + 3)(3x − 1)
A) 2x • 3x + 3 • 3x
B) 2x • 3x + 3x • -1
C) 2x • 3x + 2x • -1 + 3 • 3x + 3 • -1
D) 2x • 3x - 2x • -1 - 3 • 3x • -1

Answer 1

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:

1. **First term:** \(2x \cdot 3x = 6x^2\)
2. **Outer term:** \(2x \cdot -1 = -2x\)
3. **Inner term:** \(3 \cdot 3x = 9x\)
4. **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.