\[\left( x + \frac{x + 1}{x} \right)^{2} - \left( x - \frac{x + 1}{x} \right)^{2} =\]
\[= \left( x + \frac{x + 1}{x} - x + \frac{x + 1}{x} \right)\left( x + \frac{x + 1}{x} + x - \frac{x + 1}{x} \right) =\]
\[= \frac{x + 1 + x + 1}{x} \cdot 2x = (2x + 2) \cdot 2 =\]
\[= 4x + 4.\]