Từ \(a+b+c=0\Rightarrow b+c=-a\)

\(\Rightarrow\left(b+c\right)^2=\left(-a\right)^2\Rightarrow b^2+2bc+c^2=a^2\)

\(\Rightarrow b^2+c^2-a^2=-2bc\)\(\Rightarrow\left(b^2+c^2-a^2\right)^2=\left(-2bc\right)^2\)

\(\Rightarrow b^4+c^4+a^4+2b^2c^2-2a^2b^2-2a^2c^2=4b^2c^2\)

\(\Rightarrow a^4+b^4+c^4=2a^2b^2+2b^2c^2+2c^2a^2\)

\(\Rightarrow2(a^4+b^4+c^4)=a^4+b^4+c^4+2a^2b^2+2b^2c^2+2c^2a^2\)

\(\Rightarrow2(a^4+b^4+c^4)=(a^2+b^2+c^2)^2\)

Từ a+b+c=0⇒b+c=−a

⇒(b+c)2=(−a)2⇒b2+2bc+c2=a2

⇒b2+c2−a2=−2bc

⇒(b2+c2−a2)2=(−2bc)2

⇒b4+c4+a4+2b2c2−2a2b2−2a2c2=4b2c2

⇒a4+b4+c4=2a2b2+2b2c2+2c2a2

⇒2(a4+b4+c4)=a4+b4+c4+2a2b2+2b2c2+2c2a2

⇒2(a4+b4+c4)=(a2+b2+c2)2