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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 expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we can follow these steps:

1. Multiply the expressions step-by-step:

Start by multiplying the first two expressions:
[tex]\((7x^2) \cdot (2x^3 + 5)\)[/tex].

- Distribute [tex]\(7x^2\)[/tex] to each term in the second expression:
- [tex]\(7x^2 \cdot 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]

So, [tex]\((7x^2)(2x^3 + 5)\)[/tex] simplifies to [tex]\(14x^5 + 35x^2\)[/tex].

2. Multiply the result with the third expression:

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

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

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]

3. Combine all the terms:

Combine and organize all the terms from the distribution:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

This results in the polynomial:

[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 expressions.