Có: \(a+b+c=0\Rightarrow\left(a+b+c\right)^2=0\)
\(\Rightarrow a^2+b^2+c^2+2ab+2ac+2bc=0\)
\(\Rightarrow10+2\left(ab+ac+bc\right)=0\)
\(\Rightarrow ab+ac+bc=-5\)
\(\Rightarrow\left(ab+ac+bc\right)^2=25\)
\(\Rightarrow a^2b^2+b^2c^2+a^2c^2+2a^2bc+2ab^2c+2c^2ab=25\)
\(\Rightarrow a^2b^2+b^2c^2+a^2c^2+2abc\left(a+b+c\right)=25\)
\(\Rightarrow a^2b^2+b^2c^2+a^2c^2+2abc.0=25\)
\(\Rightarrow a^2b^2+b^2c^2+a^2c^2=25\)
Lại có: \(\left(a^2+b^2+c^2\right)^2=a^4+b^4+c^4+2a^2b^2+2b^2c^2+2a^2c^2\)
\(\Rightarrow10^2=a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+a^2c^2\right)\)
\(\Rightarrow100=a^4+b^4+c^4+2.25\)
\(\Rightarrow100=a^4+b^4+c^4+50\)
\(\Rightarrow S=a^4+b^4+c^4=50\)
