`(x^2-1)/(x-1)=-2`
What are all values of x that satisfy the equation above?

Options

A. `-3`
B. 0
C. 1
D. `-3` and `-1`

Explanation

To solve the equation \(\frac{x^2 - 1}{x - 1} = -2\), we will follow these steps:

### Step 1: Simplify the left-hand side
The expression \(\frac{x^2 - 1}{x - 1}\) can be simplified. Notice that \(x^2 - 1\) is a difference of squares, which can be factored:
\[ x^2 - 1 = (x - 1)(x + 1) \]
Thus, we can rewrite the left-hand side:
\[ \frac{(x - 1)(x + 1)}{x - 1} \]
For \(x \neq 1\), this simplifies to:
\[ x + 1 \]

### Step 2: Set the simplified expression equal to -2
Now we set the simplified expression equal to -2:
\[ x + 1 = -2 \]

### Step 3: Solve for x
To find \(x\), we subtract 1 from both sides:
\[ x = -2 - 1 \]
\[ x = -3 \]

### Step 4: Check for restrictions
We must remember that our simplification is valid only for \(x \neq 1\). Since \(x = -3\) does not violate this restriction, it is a valid solution.

### Conclusion
The only value of \(x\) that satisfies the equation is:
\[ \boxed{-3} \]

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