`1/(x + 1) - 1/(x - 1) - (x^2)/(x + 1) + (x^2)/(x -1)`, when simplified is equal to :

Options

A. 0
B. 1
C. 2
D. `-2`

Answer

To simplify the expression \( \frac{1}{x + 1} - \frac{1}{x - 1} - \frac{x^2}{x + 1} + \frac{x^2}{x - 1} \), we can follow these steps:

### Step 1: Find a common denominator
The common denominator for the fractions \( \frac{1}{x + 1} \), \( \frac{1}{x - 1} \), \( \frac{x^2}{x + 1} \), and \( \frac{x^2}{x - 1} \) is \( (x + 1)(x - 1) \).

### Step 2: Rewrite each term with the common denominator
Rewriting each term:
- \( \frac{1}{x + 1} = \frac{(x - 1)}{(x + 1)(x - 1)} \)
- \( \frac{1}{x - 1} = \frac{(x + 1)}{(x + 1)(x - 1)} \)
- \( \frac{x^2}{x + 1} = \frac{x^2(x - 1)}{(x + 1)(x - 1)} \)
- \( \frac{x^2}{x - 1} = \frac{x^2(x + 1)}{(x + 1)(x - 1)} \)

Now, substituting these into the expression, we have:
\[ \frac{(x - 1)}{(x + 1)(x - 1)} - \frac{(x + 1)}{(x + 1)(x - 1)} - \frac{x^2(x - 1)}{(x + 1)(x - 1)} + \frac{x^2(x + 1)}{(x + 1)(x - 1)} \]

### Step 3: Combine the fractions
Now we can combine the fractions:
\[ \frac{(x - 1) - (x + 1) - x^2(x - 1) + x^2(x + 1)}{(x + 1)(x - 1)} \]

### Step 4: Simplify the numerator
Now simplify the numerator:
\[ (x - 1) - (x + 1) = x - 1 - x - 1 = -2 \]
And,
\[ -x^2(x - 1) + x^2(x + 1) = -x^3 + x^2 + x^3 = x^2 \]
Combining these, we have:
\[ -2 + x^2 \]

### Step 5: Write the final expression
Thus, the expression becomes:
\[ \frac{x^2 - 2}{(x + 1)(x - 1)} \]

### Step 6: Factor the numerator
The numerator \( x^2 - 2 \) can be factored as \( (x - \sqrt{2})(x + \sqrt{2}) \).

### Final Result
The simplified expression is:
\[ \frac{x^2 - 2}{(x + 1)(x - 1)} \]