Factor `x^4 + x^2 + 1`

`x^4 + x^2 + 1` is a trinomial, that is, it consists of three terms separated by addition/subtraction signs.

It cannot be factored by the method of splitting the middle term because you can't find two numbers which multiply to give 1 and add up to give 1.

So you can try an approach similar to completing the square. Add and subtract `2x^2` from the expression,

`x^4 + x^2 + 1 + 2x^2 - 2x^2`

Rearrange the terms to bring x^4, -2x^2 and ` together,

`x^4 - 2x^2 + 1 + x^2 - 2x^2`

Factor the first three terms of the expression by comparing them with `(a - b)^2 = a^2 - 2ab + b^2` (Since `x^4 - 2x^2 + 1` can be written as `(x^2)^2 - 2x^2 + 1^2`, so it is corresponding with the right side of the formula `(a - b)^2 = a^2 - 2ab + b^2`)

`(x^2 - 1^2)^2 + x^2 - 2x^2`

Simplify,

`(x^2 - 1)^2 - x^2`

Now there is a difference of two squares, the first squared term being (x^2 - 1) and the second squared term being x. Applying the difference of squares formula,

`(x^2 - 1 + x)(x^2 - 1 - x)`

Thus the expression `x^4 + x^2 + 1` is completely factored to `(x^2 - 1 + x)(x^2 - 1 - x)`