High School

What is the product of the 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]

Answer :

To find the product [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex], we'll multiply the expressions step-by-step.

1. Multiply the first two expressions:

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

- Distribute [tex]\(7x^2\)[/tex] across each term inside the parentheses:

- [tex]\(7x^2 \cdot 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]

So, the first two expressions combine to:

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

2. Now multiply the result by the third expression:

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

- Distribute each term from [tex]\((14x^5 + 35x^2)\)[/tex] across each term 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 these products:

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

So, the product of the given expressions is:

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

This matches one of the answer choices: [tex]\(14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\)[/tex].

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