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------------------------------------------------ Which of the following represents the difference of the polynomials below?

\[ 7x^9 + 9x^5 + 15 - \left(10x^9 + 8x^5 + 12\right) \]

A. $-3x^9 + x^6 + 3$

B. $17x^6 + x^8 + 27$

C. $-3x^9 + 17x^5 + 27$

D. $3x^9 + 17x^5 + 27$

Let's find the difference of the two polynomials step by step:

Given polynomials:
- Polynomial 1: [tex]$7x^9 + 9x^5 + 15$[/tex]
- Polynomial 2: [tex]$(10x^9 + 8x^5 + 12)$[/tex]

To find the difference, we subtract the second polynomial from the first one:

[tex]\[
(7x^9 + 9x^5 + 15) - (10x^9 + 8x^5 + 12)
\][/tex]

Now, let's distribute the negative sign in front of the second polynomial:

[tex]\[
= 7x^9 + 9x^5 + 15 - 10x^9 - 8x^5 - 12
\][/tex]

Next, combine like terms:

- For the [tex]$x^9$[/tex] terms: [tex]$7x^9 - 10x^9 = -3x^9$[/tex]
- For the [tex]$x^5$[/tex] terms: [tex]$9x^5 - 8x^5 = 1x^5$[/tex] or simply [tex]$x^5$[/tex]
- Constant terms: [tex]$15 - 12 = 3$[/tex]

Putting it all together, the difference is:

[tex]\[
-3x^9 + x^5 + 3
\][/tex]

After analyzing the options provided:

- A. [tex]$-3x^9 + x^6 + 3$[/tex]
- B. [tex]$17x^6 + x^8 + 27$[/tex]
- C. [tex]$-3x^9 + 17x^5 + 27$[/tex]
- D. [tex]$3x^9 + 17x^5 + 27$[/tex]

We can see that our result matches none of the options exactly due to typing errors but matches most closely with:

A. [tex]$-3x^9 + x^5 + 3$[/tex]

So, the closest correct answer is option A.