Answer :
Sure! Let's find the product of the given expressions step-by-step:
The expressions given are:
[tex]\[
(7x^2), \quad (2x^3 + 5), \quad \text{and} \quad (x^2 - 4x - 9)
\][/tex]
We need to multiply these expressions together to find their product. Here's how we can proceed:
Step 1: Distribute and multiply the first two expressions
First, multiply [tex]\( 7x^2 \)[/tex] by each term inside the second expression [tex]\( (2x^3 + 5) \)[/tex]:
[tex]\[
7x^2 \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Simplifying each term, we get:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
Hence, the product of the first two expressions is:
[tex]\[
14x^5 + 35x^2
\][/tex]
Step 2: Multiply the resulting expression by the third expression
Next, we need to multiply [tex]\( (14x^5 + 35x^2) \)[/tex] by the third expression [tex]\( (x^2 - 4x - 9) \)[/tex]:
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\][/tex]
Distribute each term from the first expression to each term of the second expression:
[tex]\[
14x^5 \cdot (x^2 - 4x - 9) + 35x^2 \cdot (x^2 - 4x - 9)
\][/tex]
Now, let's handle these multiplications separately:
[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]
Therefore, the product of [tex]\( 14x^5 \)[/tex] with [tex]\( x^2 - 4x - 9 \)[/tex] is:
[tex]\[
14x^7 - 56x^6 - 126x^5
\][/tex]
Next, perform the similar operation for [tex]\( 35x^2 \)[/tex]:
[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]
Therefore, the product of [tex]\( 35x^2 \)[/tex] with [tex]\( x^2 - 4x - 9 \)[/tex] is:
[tex]\[
35x^4 - 140x^3 - 315x^2
\][/tex]
Step 3: Combine all the terms
Add up all the results from the individual terms:
[tex]\[
(14x^7 - 56x^6 - 126x^5) + (35x^4 - 140x^3 - 315x^2)
\][/tex]
This yields:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Therefore, the final product of the given expressions is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
From the options given, this matches the third option:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
The expressions given are:
[tex]\[
(7x^2), \quad (2x^3 + 5), \quad \text{and} \quad (x^2 - 4x - 9)
\][/tex]
We need to multiply these expressions together to find their product. Here's how we can proceed:
Step 1: Distribute and multiply the first two expressions
First, multiply [tex]\( 7x^2 \)[/tex] by each term inside the second expression [tex]\( (2x^3 + 5) \)[/tex]:
[tex]\[
7x^2 \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Simplifying each term, we get:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]
Hence, the product of the first two expressions is:
[tex]\[
14x^5 + 35x^2
\][/tex]
Step 2: Multiply the resulting expression by the third expression
Next, we need to multiply [tex]\( (14x^5 + 35x^2) \)[/tex] by the third expression [tex]\( (x^2 - 4x - 9) \)[/tex]:
[tex]\[
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
\][/tex]
Distribute each term from the first expression to each term of the second expression:
[tex]\[
14x^5 \cdot (x^2 - 4x - 9) + 35x^2 \cdot (x^2 - 4x - 9)
\][/tex]
Now, let's handle these multiplications separately:
[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]
Therefore, the product of [tex]\( 14x^5 \)[/tex] with [tex]\( x^2 - 4x - 9 \)[/tex] is:
[tex]\[
14x^7 - 56x^6 - 126x^5
\][/tex]
Next, perform the similar operation for [tex]\( 35x^2 \)[/tex]:
[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]
Therefore, the product of [tex]\( 35x^2 \)[/tex] with [tex]\( x^2 - 4x - 9 \)[/tex] is:
[tex]\[
35x^4 - 140x^3 - 315x^2
\][/tex]
Step 3: Combine all the terms
Add up all the results from the individual terms:
[tex]\[
(14x^7 - 56x^6 - 126x^5) + (35x^4 - 140x^3 - 315x^2)
\][/tex]
This yields:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Therefore, the final product of the given expressions is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
From the options given, this matches the third option:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
