Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 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]

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.