proof using Cauchy-Schwarz inequality 5

Question

In the proof i understood the first line but i dont know what happens suddenly on the second line.Please explain this.Thank you

Answer 1

If we perform the multiplication on the summands, we get $$a(5a+b+9c)=5a^2 +ab+9ca$$ Now we add them cyclically: $$\sum_{cyc} a(5a+b+9c) \ =\ 5a^2 +5b^2 +5c^2 +10ab+10bc+10ca\ =\ 5(a+b+c)^2 $$