Multiply the following polynomials vertically:

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

Choose the correct product:

A. [tex]\[ 2x^6 - 7x^5 - 2x^4 - 8x^3 \][/tex]

B. [tex]\[ 2x^6 + 7x^5 + 2x^4 - 8x^3 \][/tex]

C. [tex]\[ 2x^6 - 7x^5 - 2x^4 + 8x^3 \][/tex]

D. [tex]\[ 2x^6 + 7x^5 + 2x^4 + 8x^3 \][/tex]

Sure! Let's go through multiplying the polynomials [tex]$x^3 + 2x^2$[/tex] and [tex]$2x^3 + 3x^2 - 4x$[/tex] step-by-step vertically.

1. Write down the polynomials:
- First polynomial: [tex]$x^3 + 2x^2$[/tex]
- Second polynomial: [tex]$2x^3 + 3x^2 - 4x$[/tex]

2. Multiply each term of the first polynomial with each term of the second polynomial:

- Multiply [tex]$x^3$[/tex] by each term of [tex]$2x^3 + 3x^2 - 4x$[/tex]:
- [tex]$x^3 \times 2x^3 = 2x^6$[/tex]
- [tex]$x^3 \times 3x^2 = 3x^5$[/tex]
- [tex]$x^3 \times (-4x) = -4x^4$[/tex]

- Multiply [tex]$2x^2$[/tex] by each term of [tex]$2x^3 + 3x^2 - 4x$[/tex]:
- [tex]$2x^2 \times 2x^3 = 4x^5$[/tex]
- [tex]$2x^2 \times 3x^2 = 6x^4$[/tex]
- [tex]$2x^2 \times (-4x) = -8x^3$[/tex]

3. Combine like terms:

- The highest degree term is [tex]$2x^6$[/tex].
- For [tex]$x^5$[/tex] terms: [tex]$3x^5 + 4x^5 = 7x^5$[/tex].
- For [tex]$x^4$[/tex] terms: [tex]$-4x^4 + 6x^4 = 2x^4$[/tex].
- The only [tex]$x^3$[/tex] term is [tex]$-8x^3$[/tex].

4. Write the final polynomial:

[tex]\[
2x^6 + 7x^5 + 2x^4 - 8x^3
\][/tex]

Now, based on the calculations above, the answer matches option b: [tex]$2x^6 + 7x^5 + 2x^4 - 8x^3$[/tex].