## Task A-1.4.

For real numbers $a, b$ and $c$, it is given that $a+b+c=0$ and $abc=4$. Determine the value of the expression $a^{3}+b^{3}+c^{3}$.
## First Solution.

From the first equality, we have $a+b=-c$, or $(a+b)^{3}=-c^{3}$. 1 point

By applying the formula for the sum of cubes, we get

$$
\begin{array}{rlrl}
(a+b)^{3} & =a^{3}+3 a^{2} b+3 a b^{2}+b^{3}=-c^{3} & 1 \text { point } \\
a^{3} & +3 a b(a+b)+b^{3}=-c^{3} & 1 \text { point } \\
a^{3} & +b^{3}+c^{3}=-3 a b(a+b) . & 1 \text { point }
\end{array}
$$

From $a+b=-c$, we have $a b(a+b)=-a b c=-4$.

Therefore, we finally have $a^{3}+b^{3}+c^{3}=-3 a b(a+b)=(-3) \cdot(-4)=12$.