4. 2 .

Construct the function $f(x)=x^{3}+2 x$.
According to the problem, we know that $f(\alpha-1)=-2, f(\beta-1)=2$.
By the odd-even property and monotonicity of $f(x)$, we get
$$
(\alpha-1)+(\beta-1)=0 \text {. }
$$

Therefore, $\alpha+\beta=2$.
4. If real numbers $\alpha, \beta$ satisfy:
$$
\alpha^{3}-3 \alpha^{2}+5 \alpha=1, \beta^{3}-3 \beta^{2}+5 \beta=5 \text {, }
$$

then $\alpha+\beta=$ $\qquad$ .