To solve the product of the polynomials [tex]\( (d - 9) \)[/tex] and [tex]\( (2d^2 + 11d - 4) \)[/tex], we can follow these steps for polynomial multiplication:
1. Arrange the Polynomials:
[tex]\[
(d - 9) \times (2d^2 + 11d - 4)
\][/tex]
2. Multiply Each Term Individually:
We will multiply each term of the first polynomial by each term of the second polynomial and then combine like terms.
- Multiply [tex]\(d\)[/tex] by [tex]\(2d^2\)[/tex]:
[tex]\[
d \cdot 2d^2 = 2d^3
\][/tex]
- Multiply [tex]\(d\)[/tex] by [tex]\(11d\)[/tex]:
[tex]\[
d \cdot 11d = 11d^2
\][/tex]
- Multiply [tex]\(d\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[
d \cdot (-4) = -4d
\][/tex]
- Multiply [tex]\(-9\)[/tex] by [tex]\(2d^2\)[/tex]:
[tex]\[
-9 \cdot 2d^2 = -18d^2
\][/tex]
- Multiply [tex]\(-9\)[/tex] by [tex]\(11d\)[/tex]:
[tex]\[
-9 \cdot 11d = -99d
\][/tex]
- Multiply [tex]\(-9\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[
-9 \cdot (-4) = 36
\][/tex]
3. Combine the Like Terms:
Now we add all the products together:
[tex]\[
2d^3 + 11d^2 - 4d - 18d^2 - 99d + 36
\][/tex]
Combine the like terms for [tex]\(d^2\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[
2d^3 + (11d^2 - 18d^2) + (-4d - 99d) + 36
\][/tex]
Simplify:
- For [tex]\(d^2\)[/tex]:
[tex]\[
11d^2 - 18d^2 = -7d^2
\][/tex]
- For [tex]\(d\)[/tex]:
[tex]\[
-4d - 99d = -103d
\][/tex]
Therefore, combining all the terms we get:
[tex]\[
2d^3 - 7d^2 - 103d + 36
\][/tex]
4. Final Result:
The polynomial product of [tex]\( (d - 9) \)[/tex] and [tex]\( (2d^2 + 11d - 4) \)[/tex] is:
[tex]\[
2d^3 - 7d^2 - 103d + 36
\][/tex]
Thus, the correct answer from the options is:
[tex]\[ 2d^3 - 7d^2 - 103d + 36 \][/tex]
