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------------------------------------------------ Simplify the following expression:

[tex]
\[
\frac{4+8x^6-12x^2}{4x^2}
\]
[/tex]

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

b) [tex]\( x^6 + 2x^4 - 8x^2 \)[/tex]

c) [tex]\( x^2 + 2x^3 - 3 \)[/tex]

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

Sure, let's simplify the given expression step-by-step:

The expression is:
[tex]\[
\frac{4 + 8x^6 - 12x^2}{4x^2}
\][/tex]

1. Distribute the denominator to each term in the numerator:
[tex]\[
\frac{4}{4x^2} + \frac{8x^6}{4x^2} - \frac{12x^2}{4x^2}
\][/tex]

2. Simplify each fraction:
- For the first term:
[tex]\[
\frac{4}{4x^2} = \frac{4}{4} \cdot \frac{1}{x^2} = 1 \cdot \frac{1}{x^2} = x^{-2}
\][/tex]
- For the second term:
[tex]\[
\frac{8x^6}{4x^2} = \frac{8}{4} \cdot \frac{x^6}{x^2} = 2 \cdot x^{6-2} = 2x^4
\][/tex]
- For the third term:
[tex]\[
\frac{12x^2}{4x^2} = \frac{12}{4} \cdot \frac{x^2}{x^2} = 3 \cdot 1 = 3
\][/tex]

3. Combine the simplified terms:
[tex]\[
x^{-2} + 2x^4 - 3
\][/tex]

So, the simplified form of the given expression is:
[tex]\[
2x^4 - 3 + x^{-2}
\][/tex]

Among the given options, this matches with the correctly formatted answer. Therefore, the correct choice is:
```
2x4 - 3 + x(-2)
```

Therefore, the correct answer is:
```
2x^4 - 3 + x^(-2)
```