Find the product of `(x^(2) -1)/(x^(2) +1)` and `(x+2)/(x+1)`

Options

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

Answer

To find the product of the expressions \(\frac{x^2 - 1}{x^2 + 1}\) and \(\frac{x + 2}{x + 1}\), we will follow these steps:

### Step 1: Write the product of the two fractions
We start by writing the product of the two given fractions:
\[ \frac{x^2 - 1}{x^2 + 1} \cdot \frac{x + 2}{x + 1} \]

### Step 2: Factor the numerator of the first fraction
The expression \(x^2 - 1\) can be factored using the difference of squares:
\[ x^2 - 1 = (x - 1)(x + 1) \]
So, we can rewrite the product as:
\[ \frac{(x - 1)(x + 1)}{x^2 + 1} \cdot \frac{x + 2}{x + 1} \]

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

### Step 4: Cancel common factors
Notice that \(x + 1\) appears in both the numerator and the denominator, so we can cancel it out:
\[ \frac{(x - 1)(x + 2)}{x^2 + 1} \]

### Step 5: Expand the numerator
Now, we can expand the numerator:
\[ (x - 1)(x + 2) = x^2 + 2x - x - 2 = x^2 + x - 2 \]
Thus, we have:
\[ \frac{x^2 + x - 2}{x^2 + 1} \]

### Final Answer
The final result of the product is:
\[ \frac{x^2 + x - 2}{x^2 + 1} \]

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