Answer :
We begin with the expression
[tex]$$
\left(7x^2\right) \left(2x^3 + 5\right) \left(x^2 - 4x - 9\right).
$$[/tex]
Step 1. Expand the middle two factors
First, multiply
[tex]$$
(2x^3 + 5)(x^2 - 4x - 9).
$$[/tex]
Multiply each term in the first factor by each term in the second factor:
- Multiply [tex]$2x^3$[/tex] by [tex]$x^2$[/tex]:
[tex]$$
2x^3 \cdot x^2 = 2x^5.
$$[/tex]
- Multiply [tex]$2x^3$[/tex] by [tex]$-4x$[/tex]:
[tex]$$
2x^3 \cdot (-4x) = -8x^4.
$$[/tex]
- Multiply [tex]$2x^3$[/tex] by [tex]$-9$[/tex]:
[tex]$$
2x^3 \cdot (-9) = -18x^3.
$$[/tex]
- Multiply [tex]$5$[/tex] by [tex]$x^2$[/tex]:
[tex]$$
5 \cdot x^2 = 5x^2.
$$[/tex]
- Multiply [tex]$5$[/tex] by [tex]$-4x$[/tex]:
[tex]$$
5 \cdot (-4x) = -20x.
$$[/tex]
- Multiply [tex]$5$[/tex] by [tex]$-9$[/tex]:
[tex]$$
5 \cdot (-9) = -45.
$$[/tex]
Now, combine all these terms:
[tex]$$
2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45.
$$[/tex]
Step 2. Multiply the result by the first factor
Next, multiply the intermediate result by [tex]$7x^2$[/tex]:
[tex]$$
7x^2 \cdot \left(2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45\right).
$$[/tex]
Multiply [tex]$7x^2$[/tex] with each term:
- Multiply [tex]$7x^2$[/tex] by [tex]$2x^5$[/tex]:
[tex]$$
7x^2 \cdot 2x^5 = 14x^7.
$$[/tex]
- Multiply [tex]$7x^2$[/tex] by [tex]$-8x^4$[/tex]:
[tex]$$
7x^2 \cdot (-8x^4) = -56x^6.
$$[/tex]
- Multiply [tex]$7x^2$[/tex] by [tex]$-18x^3$[/tex]:
[tex]$$
7x^2 \cdot (-18x^3) = -126x^5.
$$[/tex]
- Multiply [tex]$7x^2$[/tex] by [tex]$5x^2$[/tex]:
[tex]$$
7x^2 \cdot 5x^2 = 35x^4.
$$[/tex]
- Multiply [tex]$7x^2$[/tex] by [tex]$-20x$[/tex]:
[tex]$$
7x^2 \cdot (-20x) = -140x^3.
$$[/tex]
- Multiply [tex]$7x^2$[/tex] by [tex]$-45$[/tex]:
[tex]$$
7x^2 \cdot (-45) = -315x^2.
$$[/tex]
Step 3. Write the final result
Hence, the final expanded product is
[tex]$$
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}.
$$[/tex]
[tex]$$
\left(7x^2\right) \left(2x^3 + 5\right) \left(x^2 - 4x - 9\right).
$$[/tex]
Step 1. Expand the middle two factors
First, multiply
[tex]$$
(2x^3 + 5)(x^2 - 4x - 9).
$$[/tex]
Multiply each term in the first factor by each term in the second factor:
- Multiply [tex]$2x^3$[/tex] by [tex]$x^2$[/tex]:
[tex]$$
2x^3 \cdot x^2 = 2x^5.
$$[/tex]
- Multiply [tex]$2x^3$[/tex] by [tex]$-4x$[/tex]:
[tex]$$
2x^3 \cdot (-4x) = -8x^4.
$$[/tex]
- Multiply [tex]$2x^3$[/tex] by [tex]$-9$[/tex]:
[tex]$$
2x^3 \cdot (-9) = -18x^3.
$$[/tex]
- Multiply [tex]$5$[/tex] by [tex]$x^2$[/tex]:
[tex]$$
5 \cdot x^2 = 5x^2.
$$[/tex]
- Multiply [tex]$5$[/tex] by [tex]$-4x$[/tex]:
[tex]$$
5 \cdot (-4x) = -20x.
$$[/tex]
- Multiply [tex]$5$[/tex] by [tex]$-9$[/tex]:
[tex]$$
5 \cdot (-9) = -45.
$$[/tex]
Now, combine all these terms:
[tex]$$
2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45.
$$[/tex]
Step 2. Multiply the result by the first factor
Next, multiply the intermediate result by [tex]$7x^2$[/tex]:
[tex]$$
7x^2 \cdot \left(2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45\right).
$$[/tex]
Multiply [tex]$7x^2$[/tex] with each term:
- Multiply [tex]$7x^2$[/tex] by [tex]$2x^5$[/tex]:
[tex]$$
7x^2 \cdot 2x^5 = 14x^7.
$$[/tex]
- Multiply [tex]$7x^2$[/tex] by [tex]$-8x^4$[/tex]:
[tex]$$
7x^2 \cdot (-8x^4) = -56x^6.
$$[/tex]
- Multiply [tex]$7x^2$[/tex] by [tex]$-18x^3$[/tex]:
[tex]$$
7x^2 \cdot (-18x^3) = -126x^5.
$$[/tex]
- Multiply [tex]$7x^2$[/tex] by [tex]$5x^2$[/tex]:
[tex]$$
7x^2 \cdot 5x^2 = 35x^4.
$$[/tex]
- Multiply [tex]$7x^2$[/tex] by [tex]$-20x$[/tex]:
[tex]$$
7x^2 \cdot (-20x) = -140x^3.
$$[/tex]
- Multiply [tex]$7x^2$[/tex] by [tex]$-45$[/tex]:
[tex]$$
7x^2 \cdot (-45) = -315x^2.
$$[/tex]
Step 3. Write the final result
Hence, the final expanded product is
[tex]$$
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}.
$$[/tex]
