Answer :
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)
```
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)
```
