The question asks for the correct expansion of [tex]\((3x - 2)(2x^2 + 5)\)[/tex]. To solve this, we'll apply the distributive property (also known as the FOIL method for binomials).
The distributive property states that [tex]\((a + b)(c + d) = ac + ad + bc + bd\)[/tex], which means we need to multiply each term in the first parenthesis by each term in the second parenthesis.
For [tex]\((3x - 2)(2x^2 + 5)\)[/tex], let's proceed step by step:
1. Multiply [tex]\(3x\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
3x \cdot 2x^2 = 6x^3
\][/tex]
2. Multiply [tex]\(3x\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
3x \cdot 5 = 15x
\][/tex]
3. Multiply [tex]\(-2\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
-2 \cdot 2x^2 = -4x^2
\][/tex]
4. Multiply [tex]\(-2\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
-2 \cdot 5 = -10
\][/tex]
Putting all these terms together, we have the expanded form:
[tex]\[
6x^3 + 15x - 4x^2 - 10
\][/tex]
Now, we need to verify which of the given statements matches this expansion process:
A. [tex]\(3x \cdot 2x^2 + (-2) \cdot 2x^2 + 2x^2 \cdot 5 + (-2) \cdot 5\)[/tex]
B. [tex]\(3x \cdot 2x^2 + 3x \cdot 5 + 2 \cdot 2x^2 + 2 \cdot 5\)[/tex]
C. [tex]\(3x \cdot 2x^2 + 3x \cdot 5 + (-2) \cdot 2x^2 + (-2) \cdot 5\)[/tex]
Comparing these options step-by-step:
- Option A incorrectly includes [tex]\(2x^2 \cdot 5\)[/tex].
- Option B incorrectly uses [tex]\(2\)[/tex] instead of [tex]\(-2\)[/tex].
- Option C correctly matches each term.
Thus, the correct option is:
[tex]\[
\boxed{\text{C}}
\][/tex]
