What is the product of the following expression?

[tex]
\[
\left(7x^2\right)\left(2x^3+5\right)\left(x^2-4x-9\right)
\]
[/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 proceed with the following steps:

1. Multiply [tex]$7x^2$[/tex] by the expression [tex]$(2x^3 + 5)$[/tex]:

- Distribute [tex]$7x^2$[/tex] to each term inside the parentheses:
[tex]\[
7x^2 \times (2x^3 + 5) = 7x^2 \times 2x^3 + 7x^2 \times 5
\][/tex]

- This gives:
[tex]\[
= 14x^5 + 35x^2
\][/tex]

2. Multiply the result by [tex]$(x^2 - 4x - 9)$[/tex]:

- Distribute each term of [tex]$(14x^5 + 35x^2)$[/tex] across each term in [tex]$(x^2 - 4x - 9)$[/tex].

Let's do this step by step:

Expanding [tex]$14x^5 \times (x^2 - 4x - 9)$[/tex]:

- [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]

The result is:
[tex]\[
14x^7 - 56x^6 - 126x^5
\][/tex]

Expanding [tex]$35x^2 \times (x^2 - 4x - 9)$[/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]

The result is:
[tex]\[
35x^4 - 140x^3 - 315x^2
\][/tex]

3. Combine all terms:

- After expanding, we collect all terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

This is the expanded form of the product and represents the final polynomial.