Express the following in the lowest terms.
`((2x^(2)+1)/(x-1)+(x-1)/(x+1))xx((x^(2)-1)/(2x)):`

Options

A. `(2x^(2)+3x)/(2)`
B. `(2x^(3)+3x^(2)-x+2)/(2x)`
C. `(2x^(3)-3x^(2)+x-2)/(2x)`
D. none of these

Answer

To express the given expression in the lowest terms, we will follow these steps:

### Step 1: Write down the expression
We start with the expression:
\[ \left(\frac{2x^2 + 1}{x - 1} + \frac{x - 1}{x + 1}\right) \cdot \frac{x^2 - 1}{2x} \]

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

### Step 3: Rewrite each fraction with the common denominator
The first fraction becomes:
\[ \frac{(2x^2 + 1)(x + 1)}{(x - 1)(x + 1)} \]
The second fraction becomes:
\[ \frac{(x - 1)(x - 1)}{(x - 1)(x + 1)} \]

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

### Step 5: Expand the numerator
Expanding the numerator:
1. For \((2x^2 + 1)(x + 1)\):
\[ 2x^3 + 2x^2 + x + 1 \]
2. For \((x - 1)^2\):
\[ x^2 - 2x + 1 \]

Now combine these:
\[ 2x^3 + 2x^2 + x + 1 + x^2 - 2x + 1 = 2x^3 + 3x^2 - x + 2 \]

### Step 6: Substitute back into the expression
Now we substitute back into our expression:
\[ \frac{2x^3 + 3x^2 - x + 2}{(x - 1)(x + 1)} \cdot \frac{x^2 - 1}{2x} \]

### Step 7: Simplify \(x^2 - 1\)
Notice that \(x^2 - 1\) can be factored as \((x - 1)(x + 1)\). Thus:
\[ \frac{(x - 1)(x + 1)}{2x} \]

### Step 8: Cancel common factors
Now we can cancel \((x - 1)(x + 1)\) in the numerator and denominator:
\[ \frac{2x^3 + 3x^2 - x + 2}{2x} \]

### Step 9: Simplify the final expression
Now we can simplify:
\[ \frac{2x^3 + 3x^2 - x + 2}{2x} = \frac{2x^3}{2x} + \frac{3x^2}{2x} - \frac{x}{2x} + \frac{2}{2x} = x^2 + \frac{3}{2}x - \frac{1}{2} + \frac{1}{x} \]

### Final Answer
Thus, the expression in the lowest terms is:
\[ x^2 + \frac{3}{2}x - \frac{1}{2} + \frac{1}{x} \]