How to find the max value of $a^2+b^2$ if $x^4+ax^3+3x^2+bx+1\ge0$ Let $f(x)=x^4+ax^3+3x^2+bx+1$, where $a,b \in \mathbb{R}$. If $f(x) \ge 0$ for all $x \in \mathbb{R}$, what is the maximum possible value of $a^2+b^2$?
I don't know how to proceed. Hints or help will be appreciated. 

$$f(x) = \frac{1}{4}\bigg[4x^4+4ax^3+12x^2+4bx+4\bigg]\geq 0$$
$$f(x)= \frac{1}{4}\bigg[(2x^2+ax)^2+(bx+2)^2+(12-a^2-b^2)x^2\bigg]\geq 0$$
So $$12-(a^2+b^2)\geq 0.$$ 
So $$a^2+b^2\leq 12$$ 
