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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 + 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], let's multiply the expressions step by step.

1. Multiply the first two terms: [tex]$7x^2$[/tex] and [tex]$(2x^3 + 5)$[/tex]:

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

- [tex]$7x^2 \cdot 2x^3 = 14x^{2+3} = 14x^5$[/tex]
- [tex]$7x^2 \cdot 5 = 35x^2$[/tex]

So, the result is [tex]$14x^5 + 35x^2$[/tex].

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

Now, take each term from [tex]$14x^5 + 35x^2$[/tex] and distribute it across [tex]$(x^2 - 4x - 9)$[/tex].

- First, multiply [tex]$14x^5$[/tex] by each term in [tex]$(x^2 - 4x - 9)$[/tex]:

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

- [tex]$14x^5 \cdot x^2 = 14x^{5+2} = 14x^7$[/tex]
- [tex]$14x^5 \cdot (-4x) = -56x^{5+1} = -56x^6$[/tex]
- [tex]$14x^5 \cdot (-9) = -126x^5$[/tex]

- Next, multiply [tex]$35x^2$[/tex] by each term in [tex]$(x^2 - 4x - 9)$[/tex]:

[tex]\[
35x^2 \cdot (x^2 - 4x - 9) = 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]

- [tex]$35x^2 \cdot x^2 = 35x^{2+2} = 35x^4$[/tex]
- [tex]$35x^2 \cdot (-4x) = -140x^{2+1} = -140x^3$[/tex]
- [tex]$35x^2 \cdot (-9) = -315x^2$[/tex]

3. Combine all the terms:

We sum all produced terms:

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

This is the expanded form of the product of the given expression.