`((x+1)/(x^(2)-1) -2/x)` expressed a rational expression is:

Options

A. `(x^(2) -2)/(x(x^(2) -1))`
B. `-(x-2)/(x(x-1))`
C. `(2(x+1))/(x^(3)-1)`
D. `1/(x(x+1))`

Answer

To express the given expression \(\frac{x+1}{x^2-1} - \frac{2}{x}\) as a rational expression, we will follow these steps:

### Step 1: Factor the Denominator
The denominator \(x^2 - 1\) can be factored using the difference of squares formula:
\[ x^2 - 1 = (x - 1)(x + 1) \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression by substituting the factored form of the denominator:
\[ \frac{x+1}{(x-1)(x+1)} - \frac{2}{x} \]

### Step 3: Find a Common Denominator
The common denominator for the two fractions is \(x(x-1)(x+1)\). We will rewrite each fraction with this common denominator:
1. For \(\frac{x+1}{(x-1)(x+1)}\):
\[ \frac{x+1}{(x-1)(x+1)} \cdot \frac{x}{x} = \frac{x(x+1)}{x(x-1)(x+1)} \]
2. For \(-\frac{2}{x}\):
\[ -\frac{2}{x} \cdot \frac{(x-1)(x+1)}{(x-1)(x+1)} = -\frac{2(x-1)(x+1)}{x(x-1)(x+1)} \]

### Step 4: Combine the Fractions
Now we can combine the two fractions:
\[ \frac{x(x+1) - 2(x-1)(x+1)}{x(x-1)(x+1)} \]

### Step 5: Simplify the Numerator
Now we simplify the numerator:
1. Expand \(2(x-1)(x+1)\):
\[ 2(x^2 - 1) = 2x^2 - 2 \]
2. Substitute back into the numerator:
\[ x(x+1) - (2x^2 - 2) = x^2 + x - 2x^2 + 2 = -x^2 + x + 2 \]

### Step 6: Final Expression
Now we have:
\[ \frac{-x^2 + x + 2}{x(x-1)(x+1)} \]
This can also be written as:
\[ \frac{-(x^2 - x - 2)}{x(x-1)(x+1)} \]

### Step 7: Factor the Numerator (if possible)
The numerator \(-x^2 + x + 2\) can be factored. We look for two numbers that multiply to \(-2\) and add to \(1\):
\[ -x^2 + x + 2 = -(x^2 - x - 2) = -(x-2)(x+1) \]

Thus, the final expression is:
\[ \frac{-(x-2)(x+1)}{x(x-1)(x+1)} \]

### Step 8: Cancel Common Factors
We can cancel the \((x+1)\) term:
\[ \frac{-(x-2)}{x(x-1)} \quad \text{for } x \neq -1 \]

### Final Answer
The rational expression is:
\[ \frac{-(x-2)}{x(x-1)} \]