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------------------------------------------------ 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+85x^4-140x^3-315x^2[/tex]

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

To find the product of the polynomials [tex]$(7x^2)(2x^3 + 5)(x^2 - 4x - 9)$[/tex], we will follow a step-by-step approach:

1. Distribute the first polynomial: Start by expanding [tex]$7x^2$[/tex] with the second polynomial [tex]$(2x^3 + 5)$[/tex].

[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5 = 14x^5 + 35x^2
\][/tex]

2. Multiply the result with the third polynomial: Distribute every term from the expanded result above with each term in [tex]$(x^2 - 4x - 9)$[/tex].

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

3. Expand each product:

For [tex]$14x^5$[/tex]:
[tex]\[
14x^5 \cdot x^2 = 14x^7
\][/tex]
[tex]\[
14x^5 \cdot (-4x) = -56x^6
\][/tex]
[tex]\[
14x^5 \cdot (-9) = -126x^5
\][/tex]

For [tex]$35x^2$[/tex]:
[tex]\[
35x^2 \cdot x^2 = 35x^4
\][/tex]
[tex]\[
35x^2 \cdot (-4x) = -140x^3
\][/tex]
[tex]\[
35x^2 \cdot (-9) = -315x^2
\][/tex]

4. Combine all terms:

Now, put together all the results from above expansions:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

This simplified expression looks exactly like the third choice in the given options:

[tex]\[ \boxed{3: \, 14x^7 - 56x^6 - 126x^5 + 85x^4 - 140x^3 - 315x^2} \][/tex]

Thus, the answer to the problem is the third option.