To determine which of the given expressions is equal to [tex]\((3x + 2)(2x - 1)\)[/tex], we can expand the expression step-by-step. Let's do that now.
First, recall the distributive property of multiplication over addition, which states that for any numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex]:
[tex]\[
(a + b)(c + d) = ac + ad + bc + bd
\][/tex]
In this case, we have:
[tex]\[
(3x + 2)(2x - 1)
\][/tex]
Let's identify our terms:
- [tex]\(a = 3x\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(c = 2x\)[/tex]
- [tex]\(d = -1\)[/tex]
Now, let's calculate the individual products:
1. [tex]\((3x)(2x) = 6x^2\)[/tex]
2. [tex]\((3x)(-1) = -3x\)[/tex]
3. [tex]\((2)(2x) = 4x\)[/tex]
4. [tex]\((2)(-1) = -2\)[/tex]
Next, we combine these results:
[tex]\[
6x^2 + (-3x) + 4x + (-2)
\][/tex]
Combine the like terms:
[tex]\[
6x^2 + (-3x + 4x) - 2
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
-3x + 4x = x
\][/tex]
Thus, the expanded form is:
[tex]\[
6x^2 + x - 2
\][/tex]
Therefore, the expression [tex]\((3x + 2)(2x - 1)\)[/tex] simplifies to [tex]\(6x^2 + x - 2\)[/tex].
Comparing this with the given options, we see that it matches the third option:
[tex]\[
6x^2 + x - 2
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{6x^2 + x - 2}
\][/tex]
