\(a+b+c+d=0\Leftrightarrow b+c=-\left(a+d\right)\Leftrightarrow\left(b+c\right)^3=-\left(a+d\right)^3\)
\(a^3+b^3+c^3+d^3=\left(b+c\right)^3-3bc\left(b+c\right)+\left(a+d\right)^3-3ad\left(a+d\right)\)
\(=-3bc\left(b+c\right)+3ad\left(b+c\right)\)
\(=3\left(b+c\right)\left(ad-bc\right)\)