Correct Answer: Option D

Explanation

\(\left(x + \frac{1}{x} + 1\right)^2 - \left(x - \frac{1}{x} - 1\right)^2,\)

We can use the difference of squares formula:

\(a^2 - b^2 = (a - b)(a + b),\)

where 

\(a = x + \frac{1}{x} + 1 \quad \text{and} \quad b = x - \frac{1}{x} - 1.\)

Calculating \( a - b \):

\(a - b = \left(x + \frac{1}{x} + 1\right) - \left(x - \frac{1}{x} - 1\right) = x + \frac{1}{x} + 1 - x + \frac{1}{x} + 1 = 2 \cdot \frac{1}{x} + 2 = 2\left(\frac{1}{x} + 1\right).\)

Calculating \( a + b \):

\(a + b = \left(x + \frac{1}{x} + 1\right) + \left(x - \frac{1}{x} - 1\right) = x + \frac{1}{x} + 1 + x - \frac{1}{x} - 1 = 2x.\)

Substituting back into the difference of squares formula:

\(a^2 - b^2 = (a - b)(a + b) = \left(2\left(\frac{1}{x} + 1\right)\right)(2x) = 4x\left(\frac{1}{x} + 1\right).\)

Distributing \(4x\):

\(= 4x \cdot \frac{1}{x} + 4x \cdot 1 = 4 + 4x.\)

Thus, the evaluated expression is: 4 + 4x = 4(1 + x)

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