Let Z  be a complex number such that the imaginary part of Z  is non-zero and $a=Z^2+Z+1$ is real then $a$ cannot take the value greater than or equal to $\lambda$ then $4\lambda$ is

 

$$\begin{gathered} Z^2+Z+1=a \\ \Rightarrow Z^2+Z+1-a=0 \\ Z=\frac{-1\pm \sqrt{4a-3}}{2} \end{gathered}$$

$\begin{array}{l}\Rightarrow 4\mathrm{a}<3\\ \Rightarrow \mathrm{a}<\frac{3}{4} then z is non real and \\ if a\ge \frac{3}{4}then z is real which contradicts data\end{array}$

$hence a cant take values greater than or equal to 3/4$