Factorise :
`x^(4) + x^(2) +1`

Answer

To factorise the expression \( x^4 + x^2 + 1 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^4 + x^2 + 1 \]

### Step 2: Recognize a pattern
We can notice that \( x^4 + x^2 + 1 \) can be rewritten in a way that allows us to use the difference of squares. We can express \( x^4 + x^2 + 1 \) as:
\[ (x^2)^2 + (1)^2 + (x^2 \cdot 1) \]

### Step 3: Create a perfect square
We can rewrite the expression as:
\[ (x^2 + 1)^2 - (x^2) \]
This is because:
\[ (x^2 + 1)^2 = x^4 + 2x^2 + 1 \]
So, we subtract \( x^2 \) to get:
\[ x^4 + x^2 + 1 = (x^2 + 1)^2 - x^2 \]

### Step 4: Apply the difference of squares
Now we can apply the identity \( a^2 - b^2 = (a - b)(a + b) \):
Let \( a = x^2 + 1 \) and \( b = x \):
\[ (x^2 + 1 - x)(x^2 + 1 + x) \]

### Step 5: Simplify the factors
Now we simplify the two factors:
1. \( x^2 + 1 - x = x^2 - x + 1 \)
2. \( x^2 + 1 + x = x^2 + x + 1 \)

### Final Answer
Thus, the factorised form of \( x^4 + x^2 + 1 \) is:
\[ (x^2 - x + 1)(x^2 + x + 1) \]

---