SOLUTION: how to prove that (a+b)(b+c)(c+a)>8abc

I think your problem must have said 
"greater than or equal", not "greater than"
and also that a, b, and c 
are all non-negative.

"Greater than" is not true when a=b=c,
because equality holds in that case

Substitute  for both  and :



That's true because it's equivalent to:






It's also not necessarily true when 2 of them are
negative.  So I think your problem must have said 
"greater than or equal", and also that a, b, and c 
are all non-negative.

First we prove the lemma: 



Proof:
       <----because the square of a real number is 
                            never negative

   <----squaring out the left side

 <----adding  to both sides

     <----factoring the left side 

 <----taking non-negative square roots
                            of both sides

Similarly we can prove 



and



just by changing the letters in the first proof.

Therefore, multiplying the left and right sides of the
three inequalities:





Edwin