Evaluate the determinant: left|begin{array}{c | vedclass

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(D) To evaluate the determinant $\left|\begin{array}{cc}x^{2}-x+1 & x-1 \ x+1 & x+1\end{array}ight|$,we use the formula for a $2 	imes 2$ determin

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$x^{3}-x^{2}+2$

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(D) To evaluate the determinant $\left|\begin{array}{cc}x^{2}-x+1 & x-1 \ x+1 & x+1\end{array}ight|$,we use the formula for a $2 	imes 2$ determinant: $\left|\begin{array}{cc}a & b \ c & d\end{array}ight| = ad - bc$.Applying this to the given determinant:$= (x^{2}-x+1)(x+1) - (x-1)(x+1)$Expand the first term:$(x^{2}-x+1)(x+1) = x^{3} + x^{2} - x^{2} - x + x + 1 = x^{3} + 1$Expand the second term:$(x-1)(x+1) = x^{2} - 1$Subtract the two results:$= (x^{3} + 1) - (x^{2} - 1)$$= x^{3} + 1 - x^{2} + 1$$= x^{3} - x^{2} + 2$

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

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