The value of a for which $2x^2-2(2a+1)x+a(a+1)$  may have one root less than a and other root greater that a are given by

$2x^2-2(2a+1)x+a(a+1)=0$

Given one root less than a other is greater than a

Case (1)

$△>0\Rightarrow 8a^2+8a+4>0$

$\begin{array}{l}\Rightarrow 2a^2+2a+1>0\\ \therefore D<0\\ a\in R\end{array}$

Case (2)

$\begin{array}{l}f\left(a\right)<0\\ \Rightarrow a(a+1)>0\\ \therefore a<-1ora>0\end{array}$