Ta có : \(a+b+c+d=0\)
\(\Leftrightarrow a+b=-c-d\)
\(\Leftrightarrow\left(a+b\right)^3=\left(-c-d\right)^3\)
\(\Leftrightarrow a^3+b^3+3ab.\left(a+b\right)=-c^3-d^3+3cd.\left(c+d\right)\)
\(\Leftrightarrow a^3+b^3+c^3+d^3=3cd.\left(c+d\right)-3ab.\left(a+b\right)\)
\(\Leftrightarrow a^3+b^3+c^3+d^3=3.cd.\left(a+b\right)+3ab.\left(c+d\right)\)
\(\Leftrightarrow a^3+b^3+c^3+d^3=3.\left(c+d\right)\left(cd+ab\right)\)
Ta có : a+b+c+d=0
⇔a+b=−c−d
⇔(a+b)3=(−c−d)3
⇔a3+b3+3ab.(a+b)=−c3−d3+3cd.(c+d)
⇔a3+b3+c3+d3=3cd.(c+d)−3ab.(a+b)
⇔a3+b3+c3+d3=3.cd.(a+b)+3ab.(c+d)
⇔a3+b3+c3+d3=3.(c+d)(cd+ab)
