To solve the given problem of multiplying three expressions:
[tex]\[
\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)
\][/tex]
let's break it down step by step.
First, let's consider the individual expressions:
1. [tex]\( 7 x^2 \)[/tex]
2. [tex]\( 2 x^3 + 5 \)[/tex]
3. [tex]\( x^2 - 4 x - 9 \)[/tex]
We need to multiply these three together.
### Step 1: Multiply the first two expressions
We start by multiplying [tex]\( 7 x^2 \)[/tex] with [tex]\( 2 x^3 + 5 \)[/tex]:
[tex]\[
7 x^2 \cdot (2 x^3 + 5) = 7 x^2 \cdot 2 x^3 + 7 x^2 \cdot 5
\][/tex]
[tex]\[
= 14 x^5 + 35 x^2
\][/tex]
### Step 2: Multiply the result with the third expression
Next, we take [tex]\( 14 x^5 + 35 x^2 \)[/tex] and multiply it by [tex]\( x^2 - 4 x - 9 \)[/tex]:
[tex]\[
(14 x^5 + 35 x^2) \cdot (x^2 - 4 x - 9)
\][/tex]
Distribute each term in [tex]\( 14 x^5 + 35 x^2 \)[/tex] across each term in [tex]\( x^2 - 4 x - 9 \)[/tex]:
[tex]\[
= 14 x^5 \cdot x^2 + 14 x^5 \cdot (-4x) + 14 x^5 \cdot (-9)
\][/tex]
[tex]\[
+ 35 x^2 \cdot x^2 + 35 x^2 \cdot (-4x) + 35 x^2 \cdot (-9)
\][/tex]
Perform each multiplication:
[tex]\[
= 14 x^7 + (-56 x^6) + (-126 x^5)
\][/tex]
[tex]\[
+ 35 x^4 + (-140 x^3) + (-315 x^2)
\][/tex]
### Step 3: Combine like terms
Now, we combine all the terms:
[tex]\[
= 14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
### Conclusion
Therefore, the product of the given expressions is:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
The corresponding multiple-choice answer is:
[tex]\[
\boxed{14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2}
\][/tex]
