One of the factor of `x ^(4) + 4` is

Options

A. `x ^(2) +2`
B. `x ^(2) + 2x - 2`
C. `x ^(2) - 2x + 2`
D. `x ^(2) -2`

Answer

To find one of the factors of the expression \( x^4 + 4 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^4 + 4 \]
We can rewrite \( 4 \) as \( 2^2 \):
\[ x^4 + 2^2 \]

### Step 2: Recognize the sum of squares
Notice that \( x^4 + 4 \) can be viewed as a sum of squares:
\[ x^4 + 4 = (x^2)^2 + (2)^2 \]

### Step 3: Use the identity for sum of squares
We can use the identity for the sum of squares:
\[ a^2 + b^2 = (a + bi)(a - bi) \]
In our case, \( a = x^2 \) and \( b = 2 \):
\[ x^4 + 4 = (x^2 + 2i)(x^2 - 2i) \]

### Step 4: Factor further using real numbers
However, we want to find a factor in real numbers. We can rewrite \( x^4 + 4 \) using a different approach:
\[ x^4 + 4 = x^4 + 4x^2 + 4 - 4x^2 = (x^2 + 2)^2 - (2x)^2 \]
This is now in the form of a difference of squares:
\[ (a^2 - b^2) = (a + b)(a - b) \]
where \( a = x^2 + 2 \) and \( b = 2x \).

### Step 5: Apply the difference of squares
Now we can factor it:
\[ (x^2 + 2 + 2x)(x^2 + 2 - 2x) \]

### Step 6: Identify one of the factors
Thus, one of the factors of \( x^4 + 4 \) is:
\[ x^2 + 2 + 2x \]

### Conclusion
So, one of the factors of \( x^4 + 4 \) is:
\[ x^2 + 2 + 2x \]

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