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

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

Hence required positive square root

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

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