Sure, let's start by performing the polynomial multiplication [tex]\( (x^2 + 5x + 1) \times (3x^2 - 2x + 4) \)[/tex] using the vertical multiplication method.
1. First, distribute the term [tex]\( x^2 \)[/tex] over each term in [tex]\( 3x^2 - 2x + 4 \)[/tex]:
[tex]\[
x^2 \cdot 3x^2 = 3x^4
\][/tex]
[tex]\[
x^2 \cdot (-2x) = -2x^3
\][/tex]
[tex]\[
x^2 \cdot 4 = 4x^2
\][/tex]
So the result of [tex]\( x^2 \cdot (3x^2 - 2x + 4) \)[/tex] is:
[tex]\[
3x^4 - 2x^3 + 4x^2
\][/tex]
2. Next, distribute the term [tex]\( 5x \)[/tex] over each term in [tex]\( 3x^2 - 2x + 4 \)[/tex]:
[tex]\[
5x \cdot 3x^2 = 15x^3
\][/tex]
[tex]\[
5x \cdot (-2x) = -10x^2
\][/tex]
[tex]\[
5x \cdot 4 = 20x
\][/tex]
So the result of [tex]\( 5x \cdot (3x^2 - 2x + 4) \)[/tex] is:
[tex]\[
15x^3 - 10x^2 + 20x
\][/tex]
3. Finally, distribute the term [tex]\( 1 \)[/tex] over each term in [tex]\( 3x^2 - 2x + 4 \)[/tex]:
[tex]\[
1 \cdot 3x^2 = 3x^2
\][/tex]
[tex]\[
1 \cdot (-2x) = -2x
\][/tex]
[tex]\[
1 \cdot 4 = 4
\][/tex]
So the result of [tex]\( 1 \cdot (3x^2 - 2x + 4) \)[/tex] is:
[tex]\[
3x^2 - 2x + 4
\][/tex]
Now, let's add all these results together, aligning them by their degree of [tex]\( x \)[/tex]:
[tex]\[
\begin{aligned}
&\ 3x^4 \\
&- 2x^3 + 15x^3 = 13x^3 \\
&4x^2 - 10x^2 + 3x^2 = -3x^2 \\
&20x - 2x = 18x \\
&\ 4
\end{aligned}
\][/tex]
So the final result of multiplying [tex]\( x^2 + 5x + 1 \)[/tex] and [tex]\( 3x^2 - 2x + 4 \)[/tex] is:
[tex]\[
3x^4 + 13x^3 - 3x^2 + 18x + 4
\][/tex]
Therefore, the correct answer is:
D. [tex]\( 3x^4 + 13 x^3 - 3 x^2 + 18 x + 4 \)[/tex]
