If the roots of the quadratic equation $a x^2+b x+c=0$ exceed a number $k$, then $a k^2+b k+c\gt0$ if $a\gt0, b^2-4 a c \geq 0$ and sum of the roots $\gt2 k$ Therefore, if the roots of $x^2+x+a=0$ exceed a number $a$, then $a^2+a+a\gt0,1-4 a \geq 0$ and $-1\gt2 a$
$\Rightarrow a(a+2)\gt0, a \leq \frac{1}{4}$ and $a \lt -\frac{1}{2}$
$\Rightarrow a\gt0$ or $a \lt -2, a \lt \frac{1}{4}$ and $a \lt -\frac{1}{2}$
Hence $a \lt -2$.