Answer :
To find the product of the polynomials [tex]\(-3x^2 + 5\)[/tex] and [tex]\(x^3 + 3x^2 - x - 7\)[/tex], we need to use the distributive property to multiply each term of the first polynomial by each term of the second polynomial and then combine like terms. Let's go through this step by step.
### Step-by-Step Solution:
1. Distribute [tex]\(-3x^2\)[/tex] through the second polynomial:
[tex]\[
\begin{align*}
-3x^2 \cdot x^3 &= -3x^{5}, \\
-3x^2 \cdot 3x^2 &= -9x^4, \\
-3x^2 \cdot (-x) &= 3x^3, \\
-3x^2 \cdot (-7) &= 21x^2.
\end{align*}
\][/tex]
So, multiplying [tex]\(-3x^2\)[/tex] by each term gives us:
[tex]\[
-3x^5 - 9x^4 + 3x^3 + 21x^2.
\][/tex]
2. Distribute [tex]\(5\)[/tex] through the second polynomial:
[tex]\[
\begin{align*}
5 \cdot x^3 &= 5x^3, \\
5 \cdot 3x^2 &= 15x^2, \\
5 \cdot (-x) &= -5x, \\
5 \cdot (-7) &= -35.
\end{align*}
\][/tex]
So, multiplying [tex]\(5\)[/tex] by each term gives us:
[tex]\[
5x^3 + 15x^2 - 5x - 35.
\][/tex]
3. Combine all the distributed terms:
[tex]\[
\begin{align*}
-3x^5 &- 9x^4 + 3x^3 + 21x^2 \\
&+ 5x^3 + 15x^2 - 5x - 35.
\end{align*}
\][/tex]
4. Combine the like terms:
[tex]\[
\begin{align*}
-3x^5 & \text{ (no like terms)} \\
- 9x^4 & \text{ (no like terms)} \\
(3x^3 + 5x^3) &= 8x^3 \\
(21x^2 + 15x^2) &= 36x^2 \\
- 5x & \text{ (no like terms)} \\
- 35 & \text{ (no like terms)}
\end{align*}
\][/tex]
5. Final result after combining:
[tex]\[
-3x^5 - 9x^4 + 8x^3 + 36x^2 - 5x - 35.
\][/tex]
So, the product of the polynomials [tex]\(-3x^2 + 5\)[/tex] and [tex]\(x^3 + 3x^2 - x - 7\)[/tex] is:
[tex]\(-3x^5 - 9x^4 + 8x^3 + 36x^2 - 5x - 35\)[/tex].
The correct answer is:
[tex]\[
\boxed{-3x^5 - 9x^4 + 8x^3 + 36x^2 - 5x - 35}
\][/tex]
This matches option D from the given choices.
### Step-by-Step Solution:
1. Distribute [tex]\(-3x^2\)[/tex] through the second polynomial:
[tex]\[
\begin{align*}
-3x^2 \cdot x^3 &= -3x^{5}, \\
-3x^2 \cdot 3x^2 &= -9x^4, \\
-3x^2 \cdot (-x) &= 3x^3, \\
-3x^2 \cdot (-7) &= 21x^2.
\end{align*}
\][/tex]
So, multiplying [tex]\(-3x^2\)[/tex] by each term gives us:
[tex]\[
-3x^5 - 9x^4 + 3x^3 + 21x^2.
\][/tex]
2. Distribute [tex]\(5\)[/tex] through the second polynomial:
[tex]\[
\begin{align*}
5 \cdot x^3 &= 5x^3, \\
5 \cdot 3x^2 &= 15x^2, \\
5 \cdot (-x) &= -5x, \\
5 \cdot (-7) &= -35.
\end{align*}
\][/tex]
So, multiplying [tex]\(5\)[/tex] by each term gives us:
[tex]\[
5x^3 + 15x^2 - 5x - 35.
\][/tex]
3. Combine all the distributed terms:
[tex]\[
\begin{align*}
-3x^5 &- 9x^4 + 3x^3 + 21x^2 \\
&+ 5x^3 + 15x^2 - 5x - 35.
\end{align*}
\][/tex]
4. Combine the like terms:
[tex]\[
\begin{align*}
-3x^5 & \text{ (no like terms)} \\
- 9x^4 & \text{ (no like terms)} \\
(3x^3 + 5x^3) &= 8x^3 \\
(21x^2 + 15x^2) &= 36x^2 \\
- 5x & \text{ (no like terms)} \\
- 35 & \text{ (no like terms)}
\end{align*}
\][/tex]
5. Final result after combining:
[tex]\[
-3x^5 - 9x^4 + 8x^3 + 36x^2 - 5x - 35.
\][/tex]
So, the product of the polynomials [tex]\(-3x^2 + 5\)[/tex] and [tex]\(x^3 + 3x^2 - x - 7\)[/tex] is:
[tex]\(-3x^5 - 9x^4 + 8x^3 + 36x^2 - 5x - 35\)[/tex].
The correct answer is:
[tex]\[
\boxed{-3x^5 - 9x^4 + 8x^3 + 36x^2 - 5x - 35}
\][/tex]
This matches option D from the given choices.
