Answer :
To determine the product of the expressions [tex]\((7 x^2)\)[/tex], [tex]\((2 x^3 + 5)\)[/tex], and [tex]\((x^2 - 4 x - 9)\)[/tex], we can start by multiplying these expressions step-by-step.
1. Multiply [tex]\(7 x^2\)[/tex] by [tex]\((2 x^3 + 5)\)[/tex]:
[tex]\[
7 x^2 \cdot (2 x^3 + 5)
\][/tex]
Apply the distributive property:
[tex]\[
7 x^2 \cdot 2 x^3 + 7 x^2 \cdot 5
\][/tex]
Calculate each term:
[tex]\[
14 x^5 + 35 x^2
\][/tex]
So, the intermediate product is:
[tex]\[
14 x^5 + 35 x^2
\][/tex]
2. Now multiply this intermediate result by [tex]\((x^2 - 4 x - 9)\)[/tex]:
[tex]\[
(14 x^5 + 35 x^2) \cdot (x^2 - 4 x - 9)
\][/tex]
We apply the distributive property (each term in the first polynomial must be multiplied by each term in the second polynomial):
[tex]\[
14 x^5 \cdot x^2 + 14 x^5 \cdot (-4 x) + 14 x^5 \cdot (-9) + 35 x^2 \cdot x^2 + 35 x^2 \cdot (-4 x) + 35 x^2 \cdot (-9)
\][/tex]
Now, let's compute each term:
[tex]\[
14 x^5 \cdot x^2 = 14 x^7
\][/tex]
[tex]\[
14 x^5 \cdot (-4 x) = -56 x^6
\][/tex]
[tex]\[
14 x^5 \cdot (-9) = -126 x^5
\][/tex]
[tex]\[
35 x^2 \cdot x^2 = 35 x^4
\][/tex]
[tex]\[
35 x^2 \cdot (-4 x) = -140 x^3
\][/tex]
[tex]\[
35 x^2 \cdot (-9) = -315 x^2
\][/tex]
Putting it all together:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
Therefore, the product of the expressions is:
[tex]\[
\boxed{14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2}
\][/tex]
1. Multiply [tex]\(7 x^2\)[/tex] by [tex]\((2 x^3 + 5)\)[/tex]:
[tex]\[
7 x^2 \cdot (2 x^3 + 5)
\][/tex]
Apply the distributive property:
[tex]\[
7 x^2 \cdot 2 x^3 + 7 x^2 \cdot 5
\][/tex]
Calculate each term:
[tex]\[
14 x^5 + 35 x^2
\][/tex]
So, the intermediate product is:
[tex]\[
14 x^5 + 35 x^2
\][/tex]
2. Now multiply this intermediate result by [tex]\((x^2 - 4 x - 9)\)[/tex]:
[tex]\[
(14 x^5 + 35 x^2) \cdot (x^2 - 4 x - 9)
\][/tex]
We apply the distributive property (each term in the first polynomial must be multiplied by each term in the second polynomial):
[tex]\[
14 x^5 \cdot x^2 + 14 x^5 \cdot (-4 x) + 14 x^5 \cdot (-9) + 35 x^2 \cdot x^2 + 35 x^2 \cdot (-4 x) + 35 x^2 \cdot (-9)
\][/tex]
Now, let's compute each term:
[tex]\[
14 x^5 \cdot x^2 = 14 x^7
\][/tex]
[tex]\[
14 x^5 \cdot (-4 x) = -56 x^6
\][/tex]
[tex]\[
14 x^5 \cdot (-9) = -126 x^5
\][/tex]
[tex]\[
35 x^2 \cdot x^2 = 35 x^4
\][/tex]
[tex]\[
35 x^2 \cdot (-4 x) = -140 x^3
\][/tex]
[tex]\[
35 x^2 \cdot (-9) = -315 x^2
\][/tex]
Putting it all together:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
Therefore, the product of the expressions is:
[tex]\[
\boxed{14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2}
\][/tex]