Using the properties of determinants in evaluate:

= (x² – x + 1) × (x + 1) – (x + 1) × (x – 1)

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

[∵ (a – b)(a + b) = (a² – b²)]

= x³ – x² + x + x² – x + 1 – x² + 1

= x³ – x² + 2