If `(x + 5) + (1)/( (x +1))=6,` then the value of `(x +1) ^(2)+ (1)/((x +1)^2)` is :

Options

A. 2
B. 0
C. `-2`
D. 4

Answer

To solve the equation \( (x + 5) + \frac{1}{(x + 1)} = 6 \) and find the value of \( (x + 1)^2 + \frac{1}{(x + 1)^2} \), we can follow these steps:

### Step 1: Simplify the given equation
Start with the equation:
\[ (x + 5) + \frac{1}{(x + 1)} = 6 \]
Subtract \( (x + 5) \) from both sides:
\[ \frac{1}{(x + 1)} = 6 - (x + 5) \]
This simplifies to:
\[ \frac{1}{(x + 1)} = 1 - x \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiply to get rid of the fraction:
\[ 1 = (1 - x)(x + 1) \]
Expanding the right side:
\[ 1 = (1 - x)(x + 1) = 1 + x - x - x^2 = 1 - x^2 \]
Thus, we have:
\[ 1 = 1 - x^2 \]

### Step 3: Solve for \( x^2 \)
Rearranging gives:
\[ x^2 = 0 \]
This implies:
\[ x = 0 \]

### Step 4: Find \( x + 1 \)
Now, substitute \( x = 0 \) into \( x + 1 \):
\[ x + 1 = 0 + 1 = 1 \]

### Step 5: Calculate \( (x + 1)^2 + \frac{1}{(x + 1)^2} \)
Now we need to find:
\[ (x + 1)^2 + \frac{1}{(x + 1)^2} \]
Substituting \( x + 1 = 1 \):
\[ (1)^2 + \frac{1}{(1)^2} = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \( (x + 1)^2 + \frac{1}{(x + 1)^2} \) is:
\[ \boxed{2} \]