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------------------------------------------------ What is the product of the following expression?

[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]

Answer :

To find the product [tex]\((7x^2)(2x^3+5)(x^2-4x-9)\)[/tex], we proceed with the following steps:

1. Multiply [tex]\(7x^2\)[/tex] by the expression [tex]\((2x^3 + 5)\)[/tex]:

- Distribute [tex]\(7x^2\)[/tex] to each term inside the parentheses:
[tex]\[
7x^2 \times (2x^3 + 5) = 7x^2 \times 2x^3 + 7x^2 \times 5
\][/tex]

- This gives:
[tex]\[
= 14x^5 + 35x^2
\][/tex]

2. Multiply the result by [tex]\((x^2 - 4x - 9)\)[/tex]:

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

Let's do this step by step:

Expanding [tex]\(14x^5 \times (x^2 - 4x - 9)\)[/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]

The result is:
[tex]\[
14x^7 - 56x^6 - 126x^5
\][/tex]

Expanding [tex]\(35x^2 \times (x^2 - 4x - 9)\)[/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]

The result is:
[tex]\[
35x^4 - 140x^3 - 315x^2
\][/tex]

3. Combine all terms:

- After expanding, we collect all terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

This is the expanded form of the product and represents the final polynomial.