The value of of $\begin{vmatrix}x^2 -x+1&x-1\\x+1&x+1\end{vmatrix}$ is equal to

$x^3 - x^2+2$

The correct answer is Option (3) → $x^3 - x^2+2$

Given: $\left| \begin{array}{cc} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{array} \right|$

Apply the $2 \times 2$ determinant formula: $ad - bc$

Value = $(x^2 - x + 1)(x + 1) - (x - 1)(x + 1)$

$= (x^3 - x^2 + x + x^2 - x + 1) - (x^2 - 1)$

$= (x^3 + 1) - (x^2 - 1)$

$= x^3 - x^2 + 2$