To find the product of the given functions, follow these detailed steps:
Given the expressions:
[tex]\[
\left(7 x^2\right), \quad \left(2 x^3 + 5\right), \quad \left(x^2 - 4 x - 9\right)
\][/tex]
1. Multiply the first two expressions:
[tex]\[
(7x^2) \cdot (2x^3 + 5)
\][/tex]
Distribute \(7x^2\) across each term in the second expression:
[tex]\[
7x^2 \cdot 2x^3 + 7x^2 \cdot 5 = 14x^5 + 35x^2
\][/tex]
2. Multiply the resulting expression by the third expression:
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\][/tex]
Now distribute each term in the first polynomial across each term in the second polynomial:
- First, multiply \(14x^5\):
[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]
- Then, multiply \(35x^2\):
[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 the terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Therefore, the product of the given functions is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
The correct answer is:
[tex]\[
\boxed{14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2}
\][/tex]
