High School

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What is the product?

[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 of the expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we'll multiply these expressions step-by-step.

### Step 1: Multiply the first two expressions
First, we multiply [tex]\(7x^2\)[/tex] by [tex]\((2x^3 + 5)\)[/tex]:

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

Calculate each term:

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

So, the result of the multiplication of the first two expressions is:

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

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

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

Distribute each term of the first expression to each term of the second expression:

- [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]

- [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]

Combine all these terms:

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

Therefore, the product of the expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex] is:

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

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