What is the product of the following expression?

[tex]\[(7x^2)(2x^3 + 5)(x^2 - 4x - 9)\][/tex]

A. [tex]$14x^5 - x^4 - 46x^3 - 58x^2 - 20x - 45$[/tex]

B. [tex]$14x^6 - 56x^5 - 91x^4 - 140x^3 - 315x^2$[/tex]

C. [tex]$14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$[/tex]

D. [tex]$14x^{12} - 182x^6 + 35x^4 - 455x^2$[/tex]

To find the product [tex]$(7x^2)(2x^3 + 5)(x^2 - 4x - 9)$[/tex], we'll break it down step by step.

1. Multiply the first two factors:

- Start by multiplying [tex]$7x^2$[/tex] and [tex]$(2x^3 + 5)$[/tex].
- Distribute [tex]$7x^2$[/tex] over each term in [tex]$(2x^3 + 5)$[/tex]:
[tex]\[
7x^2 \times 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \times 5 = 35x^2
\][/tex]
- Combine these results to get:
[tex]\[
14x^5 + 35x^2
\][/tex]

2. Multiply the result with the third factor:

- Now multiply [tex]$(14x^5 + 35x^2)$[/tex] by [tex]$(x^2 - 4x - 9)$[/tex].
- Distribute each term of [tex]$(14x^5 + 35x^2)$[/tex] over [tex]$(x^2 - 4x - 9)$[/tex].

3. Expand Step-by-step:

[tex]\[
14x^5 \times x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \times -4x = -56x^6
\][/tex]
[tex]\[
14x^5 \times -9 = -126x^5
\][/tex]

- Next, distribute [tex]$35x^2$[/tex]:

[tex]\[
35x^2 \times x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \times -4x = -140x^3
\][/tex]
[tex]\[
35x^2 \times -9 = -315x^2
\][/tex]

4. Combine All Terms:

- Now, combine all these terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

This expression matches the second option given in the multiple-choice answers. Therefore, the product is:

[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]