4. When the real number $a \in$ $\qquad$, there does not exist a real number $x$ such that $|x+a+1|+\left|x+a^{2}-2\right|<3$.
4. $(-\infty,-2] \cup[0,1] \cup[3, \infty)$.

No solution $\Leftrightarrow\left|a^{2}-a-3\right| \geqslant 3$
$\Leftrightarrow a^{2}-a \leqslant 0$ or $a^{2}-a-6 \geqslant 0$.