Answer :
We start with the expression
[tex]$$
(7x^2)(2x^3 + 5)(x^2 - 4x - 9).
$$[/tex]
Step 1: Multiply the first two factors.
Multiply [tex]\(7x^2\)[/tex] by each term in [tex]\(2x^3 + 5\)[/tex]:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5, \quad 7x^2 \cdot 5 = 35x^2.
\][/tex]
So, the product of the first two factors is
[tex]$$
14x^5 + 35x^2.
$$[/tex]
Step 2: Multiply the resulting expression by the third factor.
Now multiply
[tex]$$
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
$$[/tex]
by distributing each term in the first polynomial to every term in the second polynomial:
- Multiply [tex]\(14x^5\)[/tex] by each term:
[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]
- Multiply [tex]\(35x^2\)[/tex] by each term:
[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 all the terms.
Putting all of these products together, we have
[tex]$$
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2.
$$[/tex]
Thus, the fully expanded product is
[tex]$$
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}.
$$[/tex]
[tex]$$
(7x^2)(2x^3 + 5)(x^2 - 4x - 9).
$$[/tex]
Step 1: Multiply the first two factors.
Multiply [tex]\(7x^2\)[/tex] by each term in [tex]\(2x^3 + 5\)[/tex]:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5, \quad 7x^2 \cdot 5 = 35x^2.
\][/tex]
So, the product of the first two factors is
[tex]$$
14x^5 + 35x^2.
$$[/tex]
Step 2: Multiply the resulting expression by the third factor.
Now multiply
[tex]$$
(14x^5 + 35x^2) \cdot (x^2 - 4x - 9)
$$[/tex]
by distributing each term in the first polynomial to every term in the second polynomial:
- Multiply [tex]\(14x^5\)[/tex] by each term:
[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]
- Multiply [tex]\(35x^2\)[/tex] by each term:
[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 all the terms.
Putting all of these products together, we have
[tex]$$
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2.
$$[/tex]
Thus, the fully expanded product is
[tex]$$
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}.
$$[/tex]