Simple math question that seems intuitive

Let $$a,b,c,d$$ be real numbers.

$$3(a^2+b^2)+2(c^2+d^2)+2(ab+cd)$$

How can one show that this is always greater than zero, unless $$a=b=c=d=0$$.

I have to show a positivity axiom for an inner product and this is what I'm left with. Hence why i expect it to be greater than zero.

• It's not true when $a=i$ and $b=c=d=0$. The problem may be that I can't read or you have misentered the expression. Please edit to use mathjax: math.meta.stackexchange.com/questions/5020/… – Ethan Bolker Apr 1 at 17:26
• Sorry if a,b,c,d are in the reals, would this hold? I believe that may be my problem – Rick Apr 1 at 17:28
• Im expressing complex numbers as a+bi, c+di ie.) the complex numbers are R^2 equipped. So the values aboves are obviously real numbers. I’m so happy you pointed this out thank you. – Rick Apr 1 at 17:30
• You're welcome. Thanking me in a comment is not enough. You should edit the question so that it's correct and use mathjax to format the mathematics. – Ethan Bolker Apr 1 at 17:44

This is equal to $$2a^2+2b^2+c^2+d^2 +(a+b)^2+(c+d)^2$$ which is obviously positive for $$a,b,c,d\in\mathbb{R}$$ but is not necessarily so otherwise.