$\begin{array}{l} a + b + c + d = 0\\ \Leftrightarrow a + d = - \left( {b + c} \right)\\ \Leftrightarrow {\left( {a + d} \right)^3} = - {\left( {b + c} \right)^3}\\ \Leftrightarrow {a^3} + {d^3} + 3ad\left( {a + d} \right) = - \left( {{b^3} + {c^3} + 3bc\left( {b + c} \right)} \right)\\ \Leftrightarrow {a^3} + {d^3} + {b^3} + {c^3} = - 3ad\left( {a + d} \right) - 3bc\left( {b + c} \right)\\ \Leftrightarrow {a^3} + {d^3} + {b^3} + {c^3} = - 3ad\left( { - \left( {b + c} \right)} \right) - 3bc\left( {b + c} \right)\\ \Leftrightarrow {a^3} + {d^3} + {b^3} + {c^3} = 3ab\left( {b + c} \right) - 3bc\left( {b + c} \right)\\ \Leftrightarrow {a^3} + {b^3} + {c^3} + {d^3} = 3\left( {b + c} \right)\left( {ad - bc} \right) \end{array}$