Let $z$ be a complex number, then the equation $z^4+z+2=0$ cannot have a root, such that

Suppose there exists a complex number $z$ which satisfies the given equation and is such that $\left|z\right|<1.$ 

Then  $z^4+z+2=0\Rightarrow -2=z^4+z\Rightarrow |-2|=|z^4+z|$

$\Rightarrow 2\le \left|z^4\right|+\left|z\right|\Rightarrow 2<2,$ because $\left|z\right|<1$

But $2<2$ is not possible. Hence given equation cannot have a root $z$ such that $\left|z\right|<1$