Given, $\alpha$ and $\beta$ are the roots of $x^2+2 x+c=0$.
$\therefore$ Sum of roots, $\alpha+\beta=\frac{- \text { coefficient of } x}{\text { coefficient of } x^2}$
$\Rightarrow \quad \alpha+\beta=\frac{-2}{1}=-2$
and Product of roots, $\alpha \beta=\frac{\text { constant term }}{\text { coefficient of } x^2}$
$\Rightarrow \quad \alpha \beta=\frac{c}{1}=c$
Since, $\quad \alpha^3+\beta^3=4$
[given]
$$
\begin{aligned}
& \Rightarrow(\alpha+\beta)\left(\alpha^2+\beta^2-\alpha \beta\right)=4 \\
& {\left[\because a^3+b^3=(a+b)\left(a^2+b^2-a b\right)\right]} \\
& \Rightarrow(\alpha+\beta)\left[(\alpha+\beta)^2-3 \alpha \beta\right]=4 \\
& {\left[\because a^2+b^2=(a+b)^2-2 \alpha \beta\right]} \\
& \Rightarrow(-2)\left[(-2)^2-3 \times c\right]=4 \quad \text { [From Eqs. (i) and (ii)] } \\
& \Rightarrow \quad(-2)[4-3 c]=4 \\
& \Rightarrow \quad 4-3 c=\frac{-4}{2} \\
& \Rightarrow \quad 4-3 c=-2 \\
& \Rightarrow \quad-3 c=-2-4 \\
& \Rightarrow \quad-3 c=-6 \\
& c=2 \\
&
\end{aligned}
$$