Answer :
To solve the problem of finding 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 will expand it step-by-step and compare it to the provided options.
### Step-by-Step Solution
1. Expression Setup:
The given expression is:
[tex]\[
\left(7 x^2\right)\left(2 x^3 + 5\right)\left(x^2 - 4 x - 9\right)
\][/tex]
2. First Expansion:
Start by multiplying the first two terms:
[tex]\[
7 x^2 \cdot (2 x^3 + 5)
\][/tex]
Distribute [tex]\(7x^2\)[/tex]:
[tex]\[
7 x^2 \cdot 2 x^3 + 7 x^2 \cdot 5 = 14 x^5 + 35 x^2
\][/tex]
3. Next Expansion:
Now take the result from the first expansion and multiply it by the third term:
[tex]\[
(14 x^5 + 35 x^2) \cdot (x^2 - 4 x - 9)
\][/tex]
Distribute each term in the first polynomial over each term in the second polynomial:
[tex]\[
14 x^5 \cdot x^2 + 14 x^5 \cdot (-4x) + 14 x^5 \cdot (-9) + 35 x^2 \cdot x^2 + 35 x^2 \cdot (-4x) + 35 x^2 \cdot (-9)
\][/tex]
4. Perform Each Multiplication:
- [tex]\(14 x^5 \cdot x^2 = 14 x^7\)[/tex]
- [tex]\(14 x^5 \cdot (-4x) = -56 x^6\)[/tex]
- [tex]\(14 x^5 \cdot (-9) = -126 x^5\)[/tex]
- [tex]\(35 x^2 \cdot x^2 = 35 x^4\)[/tex]
- [tex]\(35 x^2 \cdot (-4x) = -140 x^3\)[/tex]
- [tex]\(35 x^2 \cdot (-9) = -315 x^2\)[/tex]
5. Combine All Terms:
Add together all the polynomial terms obtained:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
### Conclusion
The expanded expression matches one of the provided answers. Therefore, the correct product of the expression is:
[tex]\[
\boxed{14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2}
\][/tex]
### Step-by-Step Solution
1. Expression Setup:
The given expression is:
[tex]\[
\left(7 x^2\right)\left(2 x^3 + 5\right)\left(x^2 - 4 x - 9\right)
\][/tex]
2. First Expansion:
Start by multiplying the first two terms:
[tex]\[
7 x^2 \cdot (2 x^3 + 5)
\][/tex]
Distribute [tex]\(7x^2\)[/tex]:
[tex]\[
7 x^2 \cdot 2 x^3 + 7 x^2 \cdot 5 = 14 x^5 + 35 x^2
\][/tex]
3. Next Expansion:
Now take the result from the first expansion and multiply it by the third term:
[tex]\[
(14 x^5 + 35 x^2) \cdot (x^2 - 4 x - 9)
\][/tex]
Distribute each term in the first polynomial over each term in the second polynomial:
[tex]\[
14 x^5 \cdot x^2 + 14 x^5 \cdot (-4x) + 14 x^5 \cdot (-9) + 35 x^2 \cdot x^2 + 35 x^2 \cdot (-4x) + 35 x^2 \cdot (-9)
\][/tex]
4. Perform Each Multiplication:
- [tex]\(14 x^5 \cdot x^2 = 14 x^7\)[/tex]
- [tex]\(14 x^5 \cdot (-4x) = -56 x^6\)[/tex]
- [tex]\(14 x^5 \cdot (-9) = -126 x^5\)[/tex]
- [tex]\(35 x^2 \cdot x^2 = 35 x^4\)[/tex]
- [tex]\(35 x^2 \cdot (-4x) = -140 x^3\)[/tex]
- [tex]\(35 x^2 \cdot (-9) = -315 x^2\)[/tex]
5. Combine All Terms:
Add together all the polynomial terms obtained:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
### Conclusion
The expanded expression matches one of the provided answers. Therefore, the correct product of the expression is:
[tex]\[
\boxed{14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2}
\][/tex]
