Find the product of `(x^(2) - 1)/(x^(2) + 1) and (x + 3)/(x + 1)`

Options

A. `(x^(2) + 2x - 3)/(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 \(\frac{x + 3}{x + 1}\), we can follow these steps:

### Step 1: Write the product of the two fractions
We start by writing the product of the two given expressions:
\[ \frac{x^2 - 1}{x^2 + 1} \times \frac{x + 3}{x + 1} \]

### Step 2: Factor the numerator of the first fraction
Next, we can factor \(x^2 - 1\) using the difference of squares:
\[ x^2 - 1 = (x - 1)(x + 1) \]
So, we can rewrite the expression as:
\[ \frac{(x - 1)(x + 1)}{x^2 + 1} \times \frac{x + 3}{x + 1} \]

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

### Step 4: Cancel common factors
We see that \(x + 1\) is a common factor in the numerator and denominator, so we can cancel it:
\[ \frac{(x - 1)(x + 3)}{x^2 + 1} \]

### Step 5: Expand the numerator
Next, we can expand the numerator:
\[ (x - 1)(x + 3) = x^2 + 3x - x - 3 = x^2 + 2x - 3 \]
So, our expression now looks like:
\[ \frac{x^2 + 2x - 3}{x^2 + 1} \]

### Final Result
Thus, the product of the given expressions is:
\[ \frac{x^2 + 2x - 3}{x^2 + 1} \]

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