If $\alpha ,\beta$ are the roots of$x^2+2x+5=0$ then the equation whose roots are$\frac{\alpha +1}{\alpha },\frac{\beta +1}{\beta }$  is 

If $\alpha ,\beta$ are the roots of$x^2+2x+5=0$ then $\alpha +\beta =-2 ,\alpha \beta =5$

$\Rightarrow \frac{\alpha +1}{\alpha }+\frac{\beta +1}{\beta }=\frac{\alpha \beta +\beta +\alpha \beta +\alpha }{\alpha \beta }=\frac{2\alpha \beta +\alpha +\beta }{\alpha \beta }=\frac{10-2}{5}=\frac{8}{5}$

and$\Rightarrow \left(\frac{\alpha +1}{\alpha }\right)\left(\frac{\beta +1}{\beta }\right)=\frac{\alpha \beta +\alpha +\beta +1}{\alpha \beta }=\frac{5-2+1}{5}=\frac{4}{5}$

required equation is$x^2-\left(\frac{\alpha +1}{\alpha }+\frac{\beta +1}{\beta }\right)x+\left(\frac{\alpha +1}{\alpha }\frac{\beta +1}{\beta }\right)=0$

$$\begin{gathered} \Rightarrow x^2-\frac{8}{5}x+\frac{4}{5}=0 \\ \Rightarrow 5x^2-8x+4=0 \end{gathered}$$