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Answer :
To find the product [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we'll break it down step by step.
1. Multiply the first two factors:
- Start by multiplying [tex]\(7x^2\)[/tex] and [tex]\((2x^3 + 5)\)[/tex].
- Distribute [tex]\(7x^2\)[/tex] over each term in [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
7x^2 \times 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \times 5 = 35x^2
\][/tex]
- Combine these results to get:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Multiply the result with the third factor:
- Now multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
- Distribute each term of [tex]\((14x^5 + 35x^2)\)[/tex] over [tex]\((x^2 - 4x - 9)\)[/tex].
3. Expand Step-by-step:
[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]
- Next, distribute [tex]\(35x^2\)[/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]
4. Combine All Terms:
- Now, combine all these terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This expression matches the second option given in the multiple-choice answers. Therefore, the product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
1. Multiply the first two factors:
- Start by multiplying [tex]\(7x^2\)[/tex] and [tex]\((2x^3 + 5)\)[/tex].
- Distribute [tex]\(7x^2\)[/tex] over each term in [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
7x^2 \times 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \times 5 = 35x^2
\][/tex]
- Combine these results to get:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Multiply the result with the third factor:
- Now multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
- Distribute each term of [tex]\((14x^5 + 35x^2)\)[/tex] over [tex]\((x^2 - 4x - 9)\)[/tex].
3. Expand Step-by-step:
[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]
- Next, distribute [tex]\(35x^2\)[/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]
4. Combine All Terms:
- Now, combine all these terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This expression matches the second option given in the multiple-choice answers. Therefore, the product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
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