Factorize x⁴ + x² + 1.

To factorize x⁴ + x² + 1, consider the expression (x² + 1)².

We have the identity (a + b)² = a² + 2ab + b²

⇒ (x² + 1)² = x⁴ + 2x² + 1

Rearranging terms in the equation by moving one x² term to the left hand side, we have,

(x² + 1)² – x² = x⁴ + x² + 1

⇒ x⁴ + x² + 1 = (x² + 1)² – x²

Using the identity (a + b)(a – b) = a² – b², we have,

x⁴ + x² + 1 = [(x² + 1) + x][(x² + 1) – x]

∴ x⁴ + x² + 1 = (x² + x + 1) (x² – x + 1)

Hence, the factors of x⁴ + x² + 1 are (x² + x + 1) and (x² – x + 1).