Use vertical multiplication to find the product of:

[tex]
\[
\begin{array}{r}
x^3 + 2x + 3 \\
\times \quad x^3 - x + 1
\end{array}
\]
[/tex]

A. [tex] x^6 + x^4 + 4x^3 - 2x^2 - x + 3 [/tex]

B. [tex] x^6 + x^4 + 3x^3 - 2x^2 - 3x + 3 [/tex]

C. [tex] x^6 + 2x^3 - 2x^2 + 2x + 1 [/tex]

D. [tex] x^6 - x^4 + x^3 - 2x^2 + 2x + 3 [/tex]

Sure, let's find the product of the polynomials [tex]$(x^3 + 2x + 3)$[/tex] and [tex]$(x^3 - x + 1)$[/tex] using vertical multiplication. Here's the step-by-step process:

1. Write the polynomials in vertical format to organize the multiplication:

```
x^3 + 2x + 3
× x^3 - x + 1
```

2. Multiply each term in the second polynomial [tex]$(x^3 - x + 1)$[/tex] with each term in the first polynomial [tex]$(x^3 + 2x + 3)$[/tex]:

[tex]\[
\begin{array}{r}
& & x^3 & + & 2x & + & 3 \\
\times & & x^3 & - & x & + & 1 \\
\cline{2-8}
& x^6 & + & 2x^4 & + & 3x^3 \\ % x^3 * (x^3 + 2x + 3)
- & x^4 & - & 2x^2 & - & 3x \\ % -x * (x^3 + 2x + 3)
+ & x^3 & + & 2x & + & 3 \\ % 1 * (x^3 + 2x + 3)
\end{array}
\][/tex]

3. Align and add corresponding terms:

[tex]\[
\begin{array}{r}
& x^6 & + & 2x^4 & + & 3x^3 \\
- & & x^4 & - & 2x^2 & - & 3x \\
+ & & & x^3 & + & 2x & + & 3 \\
\cline{2-9}
& x^6 & + & x^4 & + & 4x^3 & - & 2x^2 & - & x & + & 3
\end{array}
\][/tex]

4. Combine like terms:
- There is only one [tex]$x^6$[/tex] term.
- For [tex]$x^4$[/tex], add [tex]$2x^4$[/tex] and [tex]$-x^4$[/tex] to get [tex]$x^4$[/tex].
- For [tex]$x^3$[/tex], add [tex]$3x^3$[/tex] and [tex]$x^3$[/tex] to get [tex]$4x^3$[/tex].
- There is only one [tex]$-2x^2$[/tex] term.
- For [tex]$x$[/tex], add [tex]$-3x$[/tex] and [tex]$2x$[/tex] to get [tex]$-x$[/tex].
- There is only one [tex]$3$[/tex] term.

So, the product is:

[tex]\[
x^6 + x^4 + 4x^3 - 2x^2 - x + 3
\][/tex]

Thus, the correct answer is:

[tex]\[
\boxed{x^6 + x^4 + 4x^3 - 2x^2 - x + 3}
\][/tex]

So, the correct option is:

A. [tex]$x^6 + x^4 + 4x^3 - 2x^2 - x + 3$[/tex]