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------------------------------------------------ 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 of the given expression [tex]$(7x^2)(2x^3 + 5)(x^2 - 4x - 9)$[/tex], we'll multiply the expressions step-by-step:

1. Multiply the first two expressions:
- Start with [tex]$7x^2$[/tex] and distribute it across [tex]$2x^3 + 5$[/tex]:
[tex]\[
7x^2 \times (2x^3 + 5) = (7x^2 \times 2x^3) + (7x^2 \times 5)
\][/tex]
- Calculate each term:
- [tex]$7x^2 \times 2x^3 = 14x^5$[/tex]
- [tex]$7x^2 \times 5 = 35x^2$[/tex]
- Combine to get: [tex]$14x^5 + 35x^2$[/tex]

2. Multiply the result with the third expression [tex]$x^2 - 4x - 9$[/tex]:
- Distribute [tex]$(14x^5 + 35x^2)$[/tex] across [tex]$(x^2 - 4x - 9)$[/tex]:
[tex]\[
(14x^5 + 35x^2) \times (x^2 - 4x - 9)
\][/tex]
- Multiply each term in the first polynomial by each term in the second polynomial:
- [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]
- [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]

3. Combine all the terms:
- Put together all the terms from the multiplication:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

The expanded form of the product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

This is the final answer for the expression [tex]$(7x^2)(2x^3 + 5)(x^2 - 4x - 9)$[/tex].