Answer :
Sure! Let's find the product of the three polynomial expressions step-by-step:
Given the expression:
[tex]\[
(7x^2)(2x^3 + 5)(x^2 - 4x - 9)
\][/tex]
We will use the distributive property of multiplication for polynomials, which involves expanding the expression.
### Step-by-Step Expansion:
1. First, multiply [tex]\(7x^2\)[/tex] with each term in the second polynomial [tex]\((2x^3 + 5):
\[
7x^2 \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\]
This gives us:
\[
14x^5 + 35x^2
\]
2. Next, multiply the result by the third polynomial \((x^2 - 4x - 9):
\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\]
3. Expand \(14x^5 \cdot (x^2 - 4x - 9):
\[
14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9)
\]
Simplifying this, we get:
\[
14x^7 - 56x^6 - 126x^5
\]
4. Now, expand \(35x^2 \cdot (x^2 - 4x - 9):
\[
35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\]
Simplifying this, we get:
\[
35x^4 - 140x^3 - 315x^2
\]
5. Finally, combine all the terms from steps 3 and 4:
\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\]
### Final Product:
\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\]
Therefore, the product of \(\left(7x^2\right)\left(2x^3 + 5\right)\left(x^2 - 4x- 9\right)\)[/tex] is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This matches one of the given options.
Given the expression:
[tex]\[
(7x^2)(2x^3 + 5)(x^2 - 4x - 9)
\][/tex]
We will use the distributive property of multiplication for polynomials, which involves expanding the expression.
### Step-by-Step Expansion:
1. First, multiply [tex]\(7x^2\)[/tex] with each term in the second polynomial [tex]\((2x^3 + 5):
\[
7x^2 \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\]
This gives us:
\[
14x^5 + 35x^2
\]
2. Next, multiply the result by the third polynomial \((x^2 - 4x - 9):
\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\]
3. Expand \(14x^5 \cdot (x^2 - 4x - 9):
\[
14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9)
\]
Simplifying this, we get:
\[
14x^7 - 56x^6 - 126x^5
\]
4. Now, expand \(35x^2 \cdot (x^2 - 4x - 9):
\[
35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\]
Simplifying this, we get:
\[
35x^4 - 140x^3 - 315x^2
\]
5. Finally, combine all the terms from steps 3 and 4:
\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\]
### Final Product:
\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\]
Therefore, the product of \(\left(7x^2\right)\left(2x^3 + 5\right)\left(x^2 - 4x- 9\right)\)[/tex] is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This matches one of the given options.
