\[
2t\sqrt {\left( {\frac{{1 + t}}{{2t}}} \right)^2  - 1}  = 2t\sqrt {\frac{{1 + 2t + t^2  - 4t^2 }}{{4t^2 }}}  = \frac{{2t}}{{2t}}\sqrt { - 3t^2  + 2t + 1}  = \sqrt { - 3t^2  + 2t + 1} 
\]