$y=f(x)=2\sin^22x+2\sin 2x+\sqrt 5$
$\begin{array}{|c|ccc|} \hline \sin 2x &-1& \qquad \dfrac{-1}{2} & \qquad1 \\ \hline & \sqrt 5& \qquad& \qquad 4+\sqrt 5 \\ f(x)& \qquad \searrow & & \nearrow \\ & &\qquad \dfrac{-1+2\sqrt 5}{2} \\ \hline \end{array} $
$\Rightarrow \min f(x)=\frac{-1+2\sqrt 5}{2}\Leftrightarrow \sin 2x=-\frac 12\Leftrightarrow \left[ \begin{array}{l} x=k\pi-\dfrac{\pi}{12}\\ x=k\pi+\frac{7}{12} \end{array} \right.$
$\max f(x)=4+\sqrt 5\Leftrightarrow \sin 2x=1\Leftrightarrow x=k\pi+\frac{\pi}4$