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------------------------------------------------ What is the sum of [tex]\left(2x^4 + 5x^3 - 8x^2 - x + 10\right)[/tex] and [tex]\left(8x^4 - 4x^3 + x^2 - x + 2\right)[/tex]?

A. [tex]10x^4 + 9x^3 - 7x^2 - 2x + 12[/tex]

B. [tex]10x^8 + x^6 - 7x^4 - 2x^2 + 12[/tex]

C. [tex]10x^4 + x^3 - 9x^2 + 12[/tex]

D. [tex]10x^4 + x^3 - 7x^2 - 2x + 12[/tex]

E. [tex]10x^4 + x^3 - 9x^2 - 2x + 12[/tex]

To find the sum of the polynomials [tex]$(2x^4 + 5x^3 - 8x^2 - x + 10)$[/tex] and [tex]$(8x^4 - 4x^3 + x^2 - x + 2)$[/tex], we simply add the corresponding coefficients from each polynomial. Let's break it down step-by-step:

1. Identify the coefficients:
- For [tex]$x^4$[/tex], we have [tex]$2$[/tex] from the first polynomial and [tex]$8$[/tex] from the second.
- For [tex]$x^3$[/tex], we have [tex]$5$[/tex] from the first polynomial and [tex]$-4$[/tex] from the second.
- For [tex]$x^2$[/tex], we have [tex]$-8$[/tex] from the first polynomial and [tex]$1$[/tex] from the second.
- For [tex]$x$[/tex], we have [tex]$-1$[/tex] from both polynomials.
- For the constant term, we have [tex]$10$[/tex] from the first polynomial and [tex]$2$[/tex] from the second.

2. Add the coefficients:
- [tex]$x^4$[/tex] term: [tex]$2 + 8 = 10$[/tex]
- [tex]$x^3$[/tex] term: [tex]$5 - 4 = 1$[/tex]
- [tex]$x^2$[/tex] term: [tex]$-8 + 1 = -7$[/tex]
- [tex]$x$[/tex] term: [tex]$-1 - 1 = -2$[/tex]
- Constant term: [tex]$10 + 2 = 12$[/tex]

3. Combine the results into the polynomial:
[tex]\[
10x^4 + 1x^3 - 7x^2 - 2x + 12
\][/tex]

The sum of the polynomials is [tex]$10x^4 + x^3 - 7x^2 - 2x + 12$[/tex].