`(x^(8)+4)/(x^(4)+2x^(2)+2)` on simplification, equals

Options

A. `x^(4)+2x^(2)-2`
B. `x^(4)-2x^(2)+2`
C. `x^(4)-2x^(2)-2`
D. cannot be simplified

Answer

To simplify the expression \(\frac{x^8 + 4}{x^4 + 2x^2 + 2}\), we can follow these steps:

### Step-by-Step Solution:

1. **Recognize the Numerator**:
The numerator is \(x^8 + 4\). We can rewrite \(4\) as \(2^2\). Thus, we can express the numerator as:
\[ x^8 + 4 = x^8 + 2^2 \]

2. **Factor the Numerator**:
We can use the sum of squares formula, which states \(a^2 + b^2 = (a + bi)(a - bi)\). However, in this case, we can also use the identity for the difference of squares:
\[ a^2 - b^2 = (a + b)(a - b) \]
We can rewrite \(x^8 + 4\) as:
\[ (x^4)^2 + (2)^2 \]
This can be factored as:
\[ (x^4 + 2i)(x^4 - 2i) \]
However, since we are working with real numbers, we will use a different approach.

3. **Rewrite the Numerator**:
Notice that we can also express \(x^8 + 4\) as:
\[ (x^4 + 2)^2 - (2x^2)^2 \]
This is a difference of squares, which can be factored as:
\[ ((x^4 + 2) + 2x^2)((x^4 + 2) - 2x^2) \]
Thus, we have:
\[ (x^4 + 2 + 2x^2)(x^4 + 2 - 2x^2) \]

4. **Factor the Denominator**:
The denominator is \(x^4 + 2x^2 + 2\). We can recognize that it can be rewritten as:
\[ x^4 + 2x^2 + 2 = (x^4 + 2) + 2x^2 \]
This doesn't factor nicely, so we will leave it as is.

5. **Simplify the Expression**:
Now, substituting back into our expression:
\[ \frac{(x^4 + 2 + 2x^2)(x^4 + 2 - 2x^2)}{x^4 + 2x^2 + 2} \]
We see that \(x^4 + 2 - 2x^2\) can be simplified with the denominator.

6. **Final Simplification**:
The expression simplifies to:
\[ x^4 + 2 - 2x^2 \]
Therefore, the final answer is:
\[ x^4 - 2x^2 + 2 \]

### Final Answer:
\[ \frac{x^8 + 4}{x^4 + 2x^2 + 2} = x^4 - 2x^2 + 2 \]