College

What is the product?

\[ \left(7x^2\right)\left(2x^3 + 5\right)\left(x^2 - 4x - 9\right) \]

A. \( 14x^5 - x^4 - 46x^3 - 58x^2 - 20x - 45 \)

B. \( 14x^6 - 56x^5 - 91x^4 - 140x^3 - 315x^2 \)

C. \( 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \)

D. \( 14x^{12} - 182x^6 + 35x^4 - 455x^2 \)

Answer :

Sure! Let's find the product [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex] step-by-step:

1. First Step: Multiply the first two expressions:
[tex]\[
(7x^2)(2x^3 + 5)
\][/tex]
Distribute [tex]\(7x^2\)[/tex] across the terms in the parentheses:
[tex]\[
7x^2 \cdot 2x^3 + 7x^2 \cdot 5 = 14x^5 + 35x^2
\][/tex]

2. Second Step: Multiply the result by the third expression:
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]

3. Third Step: Distribute [tex]\(14x^5\)[/tex] across [tex]\(x^2 - 4x - 9\)[/tex]:
[tex]\[
14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9) = 14x^7 - 56x^6 - 126x^5
\][/tex]

4. Fourth Step: Distribute [tex]\(35x^2\)[/tex] across [tex]\(x^2 - 4x - 9\)[/tex]:
[tex]\[
35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9) = 35x^4 - 140x^3 - 315x^2
\][/tex]

5. Fifth Step: Combine all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

So, the final product is:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]

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