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

Given that $\alpha ,\beta$ are the roots of the quadratic equation $x^2+2x+5=0$

${\alpha }^2+2\alpha +5=0$, it implies that ${\alpha }^2+\alpha +1=-\alpha -4$

Equation whose roots are ${\alpha }^2+\alpha +1, {\beta }^2+\beta +1$,is same as the equation whose roots are $-\alpha -4,-\beta -4$

Therefore, the required equation is 

$\begin{array}{rcl}{x}^2-x\left(-\alpha -4-\beta -4\right)+\left(-\alpha -4\right)\left(-\beta -4\right)& =& 0\\ x^2+x\left(\alpha +\beta +8\right)+\left(\alpha \beta +4\left(\alpha +\beta \right)+16\right)& =& 0\\ x^2+x\left(-2+8\right)+\left(5-8+16\right)& =& 0\\ x^2+6x+13& =& 0\\ & & \end{array}$