Find the difference:

[tex]\left(3x^6 + 7x^5 - 9x^4 + 8x^3 - 10x^2\right) - \left(5x^6 - 2x^5 - 2x + 12\right)[/tex].

A. [tex]-2x^{12} + 5x^{10} - 10x^5 + 22x^2[/tex]

B. [tex]-2x^6 + 9x^5 - 11x^3 + 2x^2 - 12[/tex]

C. [tex]8x^6 + 5x^5 - 11x^4 + 2x^3 - 2x + 12[/tex]

D. [tex]-2x^6 + 9x^5 - 9x^4 + 8x^3 - 10x^2 + 2x - 12[/tex]

E. [tex]3x^6 + 2x^5 - 9x^4 + 8x^3 - 10x^2 + 2x - 12[/tex]

To find the difference between the two polynomials, we will subtract each corresponding term in the second polynomial from the first polynomial. Here’s how we will do this step by step:

### Step 1: Identify the coefficients of both polynomials

- For the first polynomial [tex]$ 3x^6 + 7x^5 - 9x^4 + 8x^3 - 10x^2 $[/tex], the coefficients are:
- [tex]$3x^6$[/tex], [tex]$7x^5$[/tex], [tex]$-9x^4$[/tex], [tex]$8x^3$[/tex], [tex]$-10x^2$[/tex]

- For the second polynomial [tex]$ 5x^6 - 2x^5 - 2x + 12 $[/tex], the coefficients are:
- [tex]$5x^6$[/tex], [tex]$-2x^5$[/tex], [tex]$0x^4$[/tex], [tex]$0x^3$[/tex], [tex]$0x^2$[/tex], [tex]$-2x$[/tex], [tex]$12$[/tex]

### Step 2: Rewrite the polynomials to match their terms

We will ensure both polynomials have the same degree by inserting zero coefficients where necessary.

- First polynomial: [tex]$ 3x^6 + 7x^5 - 9x^4 + 8x^3 - 10x^2 + 0x + 0 $[/tex]
- Second polynomial: [tex]$ 5x^6 - 2x^5 + 0x^4 + 0x^3 + 0x^2 - 2x + 12 $[/tex]

### Step 3: Subtract corresponding coefficients

Now, subtract each coefficient in the second polynomial from the first:

- [tex]$ (3x^6 - 5x^6) = -2x^6 $[/tex]
- [tex]$ (7x^5 - (-2x^5)) = 9x^5 $[/tex]
- [tex]$ (-9x^4 - 0x^4) = -9x^4 $[/tex]
- [tex]$ (8x^3 - 0x^3) = 8x^3 $[/tex]
- [tex]$ (-10x^2 - 0x^2) = -10x^2 $[/tex]
- [tex]$ (0x - (-2x)) = 2x $[/tex]
- [tex]$ (0 - 12) = -12 $[/tex]

### Step 4: Write the resulting polynomial

Combine the results to get the final polynomial:

[tex]\[
-2x^6 + 9x^5 - 9x^4 + 8x^3 - 10x^2 + 2x - 12
\][/tex]

Therefore, the correct choice from the given options is:

d. [tex]$-2x^6 + 9x^5 - 9x^4 + 8x^3 - 10x^2 + 2x - 12$[/tex]