Answer :
We want to multiply the three factors:
[tex]$$
7x^2,\quad 2x^3+5,\quad \text{and} \quad x^2-4x-9.
$$[/tex]
Step 1. Multiply the first two factors
Multiply
[tex]$$
7x^2 \quad \text{and} \quad (2x^3+5):
$$[/tex]
[tex]\[
7x^2 \cdot (2x^3+5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5.
\][/tex]
Compute each term:
[tex]\[
7x^2 \cdot 2x^3 = 14x^{2+3} = 14x^5,
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2.
\][/tex]
Thus, the intermediate product is:
[tex]$$
14x^5 + 35x^2.
$$[/tex]
Step 2. Multiply the result by the third factor
Now multiply:
[tex]$$
(14x^5+35x^2)(x^2-4x-9).
$$[/tex]
Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial:
1. Multiply [tex]$14x^5$[/tex] by each term in [tex]$x^2-4x-9$[/tex]:
[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]
2. Multiply [tex]$35x^2$[/tex] by each term in [tex]$x^2-4x-9$[/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]
Step 3. Combine like terms
Write all the terms together:
[tex]\[
14x^7 -56x^6 -126x^5 +35x^4 -140x^3 -315x^2.
\][/tex]
There are no like terms to combine further, so this is the complete expanded product.
Final Answer
The product of
[tex]$$\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)$$[/tex]
is:
[tex]$$
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2.
$$[/tex]
Thus, the correct option is the third one.
[tex]$$
7x^2,\quad 2x^3+5,\quad \text{and} \quad x^2-4x-9.
$$[/tex]
Step 1. Multiply the first two factors
Multiply
[tex]$$
7x^2 \quad \text{and} \quad (2x^3+5):
$$[/tex]
[tex]\[
7x^2 \cdot (2x^3+5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5.
\][/tex]
Compute each term:
[tex]\[
7x^2 \cdot 2x^3 = 14x^{2+3} = 14x^5,
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2.
\][/tex]
Thus, the intermediate product is:
[tex]$$
14x^5 + 35x^2.
$$[/tex]
Step 2. Multiply the result by the third factor
Now multiply:
[tex]$$
(14x^5+35x^2)(x^2-4x-9).
$$[/tex]
Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial:
1. Multiply [tex]$14x^5$[/tex] by each term in [tex]$x^2-4x-9$[/tex]:
[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]
2. Multiply [tex]$35x^2$[/tex] by each term in [tex]$x^2-4x-9$[/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]
Step 3. Combine like terms
Write all the terms together:
[tex]\[
14x^7 -56x^6 -126x^5 +35x^4 -140x^3 -315x^2.
\][/tex]
There are no like terms to combine further, so this is the complete expanded product.
Final Answer
The product of
[tex]$$\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)$$[/tex]
is:
[tex]$$
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2.
$$[/tex]
Thus, the correct option is the third one.
