Evaluate $\left|\begin{array}{cc}x & x+1 \\ x-1 & x\end{array}\right|$

Step 1: Recall Determinant Formula

To evaluate a $2 \times 2$ determinant, we multiply the elements on the main diagonal (top-left to bottom-right) and subtract the product of the elements on the anti-diagonal (top-right to bottom-left).

Step 2: Identify Elements

In the given determinant, we identify the values for $a, b, c,$ and $d$ by comparing it with the general form of a $2 \times 2$ determinant.

Step 3: Substitute and Simplify

Now, substitute these values into the determinant formula. We multiply $x$ by $x$, and subtract the product of $(x+1)$ and $(x-1)$.

Step 4: Expand and Combine Terms

Expand the products. The term $(x+1)(x-1)$ is a difference of squares, which simplifies to $x^2 - 1^2 = x^2 - 1$. Then, distribute the negative sign and combine like terms.

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