What is the product?

[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 solve this problem, we need to find the product of the expression:

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

Let's break it down into steps:

### Step 1: Expand the Inner Expressions

First, we'll multiply [tex]$7x^2$[/tex] with each term of [tex]$(2x^3 + 5)$[/tex] individually:

- [tex]$7x^2 \times 2x^3 = 14x^5$[/tex],
- [tex]$7x^2 \times 5 = 35x^2$[/tex].

Now our expression is:

[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9).
\][/tex]

### Step 2: Multiply the Resulting Expression

Next, distribute each term from [tex]$(14x^5 + 35x^2)$[/tex] to each term in [tex]$(x^2 - 4x - 9)$[/tex]:

- For [tex]$14x^5$[/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].

- For [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].

### Step 3: Write the Full Expansion and Combine Like Terms

Now, let's write all the results from the multiplication:

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

Look through these terms and ensure all like terms have been combined. Since all terms are distinct besides being different powers of [tex]$x$[/tex], there are no additional terms to combine.

Thus, the final expanded product of the expression is:

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

This is the correct answer to the problem.