College

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]

Answer :

To find the product of the given expression [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex], we follow these steps:

1. Distribute the first part [tex]\((7x^2)\)[/tex] to the second part [tex]\((2x^3 + 5)\)[/tex]:

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

This results in:

[tex]\[
14x^5 + 35x^2
\][/tex]

2. Now multiply this result by the third part [tex]\((x^2 - 4x - 9)\)[/tex]:

We distribute each term from [tex]\(14x^5 + 35x^2\)[/tex] across [tex]\(x^2 - 4x - 9\)[/tex].

- First, distribute [tex]\(14x^5\)[/tex]:

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

[tex]\[
= 14x^7 - 56x^6 - 126x^5
\][/tex]

- Next, distribute [tex]\(35x^2\)[/tex]:

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

[tex]\[
= 35x^4 - 140x^3 - 315x^2
\][/tex]

3. Combine all terms:

- Combine the terms from both distributions:

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

This is the expanded expression, which is the product of the given expressions:

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

This is the final answer.

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