To simplify the equation \((x-1)^2 + x^2 - 1 = (x-1)(x+3)\), we can perform the following steps:

1. **Expand both sides**:
- Left Side: \((x-1)^2 + x^2 - 1\)
- \((x-1)^2 = x^2 - 2x + 1\)
- Therefore, \(x^2 - 2x + 1 + x^2 - 1 = 2x^2 - 2x\)

- Right Side: \((x-1)(x+3)\)
- \(= x^2 + 3x - x - 3 = x^2 + 2x - 3\)

2. **Set the equation**:
\[
2x^2 - 2x = x^2 + 2x - 3
\]

3. **Rearranging the terms**:
\[
2x^2 - 2x - x^2 - 2x + 3 = 0
\]
\[
x^2 - 4x + 3 = 0
\]

4. **Factoring**:
\[
(x-3)(x-1) = 0
\]

5. **Finding the solutions**:
- \(x-3 = 0 \Rightarrow x = 3\)
- \(x-1 = 0 \Rightarrow x = 1\)

Thus, the solutions are \(x = 3\) and \(x = 1\).