Let
$$g(x)=ax^4+bx^3+cx^2+dx+e$$
be a polynomial of degree $4$ with $a>0$. Can we determine coefficients $a,b,c,d,e$ such that $g(x) \ge 0$ for all $x\in\mathbb{R}$? I'd like to make sure that my idea works. In addition to the conditions $\Delta>0$, $P>0$, and $D>0$ given in https://en.wikipedia.org/wiki/Quartic_function , do we need any other condition?
Any reference, suggestion, idea, or comment is welcome. Thank you!
Without the loss of generality we can take $a=1$. The we have $$f(x)=x^4+bx^3+cx^2+dx+e~~~~(1)$$ Let $$c \ge \frac{b^2+d^2}{4}~~~~(2)$$ Then we can re-write $f(x)$ as $$f(x)=\left(x^2+\frac{bx}{2}\right)^2+\left(c-\frac{b^2}{4}-\frac{d^2}{4}\right) x^2+\left(\frac{dx}{2}+1\right)^2+e-1.$$ So when $a=1$, $e\ge 1$ and (2) is met $f(x) \ge 0, \forall~ x\in R$