Answer :
We want to multiply the two polynomials
[tex]$$
(3x^2 - 4x + 5)(x^2 - 3x + 2).
$$[/tex]
We'll apply the distributive property, multiplying each term in the first polynomial by each term in the second polynomial, and then combine like terms.
1. Multiply the term [tex]$3x^2$[/tex] by each term in the second polynomial:
[tex]\[
\begin{aligned}
3x^2 \cdot x^2 &= 3x^4, \\
3x^2 \cdot (-3x) &= -9x^3, \\
3x^2 \cdot 2 &= 6x^2.
\end{aligned}
\][/tex]
2. Multiply the term [tex]$-4x$[/tex] by each term in the second polynomial:
[tex]\[
\begin{aligned}
-4x \cdot x^2 &= -4x^3, \\
-4x \cdot (-3x) &= 12x^2, \\
-4x \cdot 2 &= -8x.
\end{aligned}
\][/tex]
3. Multiply the term [tex]$5$[/tex] by each term in the second polynomial:
[tex]\[
\begin{aligned}
5 \cdot x^2 &= 5x^2, \\
5 \cdot (-3x) &= -15x, \\
5 \cdot 2 &= 10.
\end{aligned}
\][/tex]
4. Combine like terms:
- For the [tex]$x^4$[/tex] term:
[tex]$$
3x^4.
$$[/tex]
- For the [tex]$x^3$[/tex] terms:
[tex]$$
-9x^3 + (-4x^3) = -13x^3.
$$[/tex]
- For the [tex]$x^2$[/tex] terms:
[tex]$$
6x^2 + 12x^2 + 5x^2 = 23x^2.
$$[/tex]
- For the [tex]$x$[/tex] terms:
[tex]$$
-8x + (-15x) = -23x.
$$[/tex]
- For the constant term:
[tex]$$
10.
$$[/tex]
Thus, the product is
[tex]$$
3x^4 - 13x^3 + 23x^2 - 23x + 10.
$$[/tex]
Comparing with the given options, the correct answer is:
D. [tex]$3 x^4 - 13 x^3 + 23 x^2 - 23 x + 10$[/tex].
[tex]$$
(3x^2 - 4x + 5)(x^2 - 3x + 2).
$$[/tex]
We'll apply the distributive property, multiplying each term in the first polynomial by each term in the second polynomial, and then combine like terms.
1. Multiply the term [tex]$3x^2$[/tex] by each term in the second polynomial:
[tex]\[
\begin{aligned}
3x^2 \cdot x^2 &= 3x^4, \\
3x^2 \cdot (-3x) &= -9x^3, \\
3x^2 \cdot 2 &= 6x^2.
\end{aligned}
\][/tex]
2. Multiply the term [tex]$-4x$[/tex] by each term in the second polynomial:
[tex]\[
\begin{aligned}
-4x \cdot x^2 &= -4x^3, \\
-4x \cdot (-3x) &= 12x^2, \\
-4x \cdot 2 &= -8x.
\end{aligned}
\][/tex]
3. Multiply the term [tex]$5$[/tex] by each term in the second polynomial:
[tex]\[
\begin{aligned}
5 \cdot x^2 &= 5x^2, \\
5 \cdot (-3x) &= -15x, \\
5 \cdot 2 &= 10.
\end{aligned}
\][/tex]
4. Combine like terms:
- For the [tex]$x^4$[/tex] term:
[tex]$$
3x^4.
$$[/tex]
- For the [tex]$x^3$[/tex] terms:
[tex]$$
-9x^3 + (-4x^3) = -13x^3.
$$[/tex]
- For the [tex]$x^2$[/tex] terms:
[tex]$$
6x^2 + 12x^2 + 5x^2 = 23x^2.
$$[/tex]
- For the [tex]$x$[/tex] terms:
[tex]$$
-8x + (-15x) = -23x.
$$[/tex]
- For the constant term:
[tex]$$
10.
$$[/tex]
Thus, the product is
[tex]$$
3x^4 - 13x^3 + 23x^2 - 23x + 10.
$$[/tex]
Comparing with the given options, the correct answer is:
D. [tex]$3 x^4 - 13 x^3 + 23 x^2 - 23 x + 10$[/tex].