Factors of `x ^(4) + 5x ^(2) + 9` are

Options

A. `(x ^(2) + 2x + 3) (x ^(2) + 3x + 3)`
B. `(x ^(2) - x + 3) (x ^(2) - x -3)`
C. `(x ^(2) - x -3) (x ^(3) + x + 3)`
D. `(x ^(2) - x + 3) (x ^(2) + x + 3)`

Answer

To factor the expression \( x^4 + 5x^2 + 9 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^4 + 5x^2 + 9 \]
Notice that \( x^4 \) can be rewritten as \( (x^2)^2 \) and \( 9 \) can be rewritten as \( 3^2 \). This gives us:
\[ (x^2)^2 + 5x^2 + 3^2 \]

### Step 2: Identify a suitable form
We want to express \( 5x^2 \) in a way that allows us to use the identity \( a^2 + b^2 + 2ab \). To do this, we can express \( 5x^2 \) as:
\[ 6x^2 - x^2 \]
This allows us to rewrite the expression as:
\[ (x^2)^2 + 6x^2 - x^2 + 3^2 \]

### Step 3: Group the terms
Now we can group the terms:
\[ (x^2)^2 + 6x^2 + 3^2 - x^2 \]
This can be rearranged to:
\[ (x^2 + 3)^2 - x^2 \]

### Step 4: Apply the difference of squares formula
We can now apply the difference of squares formula, which states that \( a^2 - b^2 = (a + b)(a - b) \). Here, let \( a = x^2 + 3 \) and \( b = x \):
\[ (x^2 + 3 + x)(x^2 + 3 - x) \]

### Step 5: Write the final factors
Thus, the factors of the expression \( x^4 + 5x^2 + 9 \) are:
\[ (x^2 + x + 3)(x^2 - x + 3) \]

### Final Answer
The factors of \( x^4 + 5x^2 + 9 \) are:
\[ (x^2 + x + 3)(x^2 - x + 3) \]

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