I expanded the LHS and got a^2b+ab^2+a^2c+ac^2+b^2c+bc^2+2abc, so -2abc on both sides, so we need to prove a^2b+ab^2+a^2c+ac^2+b^2c+bc^2 is greater than or equal to 6abc.
using the a^2+b^2 >= 2ab, a^2c+b^2c >= 2abc
similarly the other two groups are >= 2abc
so the six terms are >= 6abc