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------------------------------------------------ Multiply: [tex](2x - 4)(x^2 + 3x - 7)[/tex]

A. [tex]-2x^3 - 6x^2 + 14x + 28[/tex]
B. [tex]-4x^2 - 12x + 28[/tex]
C. [tex]2x^3 + 2x^2 - 26x + 28[/tex]
D. [tex]2x^3 + 6x^2 - 14x[/tex]

To multiply the expressions [tex]\((2x - 4)\)[/tex] and [tex]\((x^2 + 3x - 7)\)[/tex], we'll use the distributive property, also known as the distributive law of multiplication over addition. Here's how you can do it step-by-step:

1. Distribute [tex]\(2x\)[/tex]:

Multiply [tex]\(2x\)[/tex] by each term inside the second expression [tex]\((x^2 + 3x - 7)\)[/tex]:

- [tex]\(2x \times x^2 = 2x^3\)[/tex]
- [tex]\(2x \times 3x = 6x^2\)[/tex]
- [tex]\(2x \times (-7) = -14x\)[/tex]

So, distributing [tex]\(2x\)[/tex] gives us: [tex]\(2x^3 + 6x^2 - 14x\)[/tex].

2. Distribute [tex]\(-4\)[/tex]:

Multiply [tex]\(-4\)[/tex] by each term inside the second expression [tex]\((x^2 + 3x - 7)\)[/tex]:

- [tex]\(-4 \times x^2 = -4x^2\)[/tex]
- [tex]\(-4 \times 3x = -12x\)[/tex]
- [tex]\(-4 \times (-7) = 28\)[/tex]

So, distributing [tex]\(-4\)[/tex] gives us: [tex]\(-4x^2 - 12x + 28\)[/tex].

3. Combine like terms:

Now, add up all the results from both distributions:

[tex]\[
2x^3 + 6x^2 - 14x - 4x^2 - 12x + 28
\][/tex]

- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(6x^2 - 4x^2 = 2x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-14x - 12x = -26x\)[/tex]

Thus, the combined and simplified expression is:

[tex]\[
2x^3 + 2x^2 - 26x + 28
\][/tex]

Therefore, the correct answer is option c: [tex]\(2x^3 + 2x^2 - 26x + 28\)[/tex].