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### 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)`