Fix a positive even integer $d$ and consider the polynomial $f(x)=c_d x^d+\ldots+c_1x+c_0$, where the $c_i$ are independent random variables that follow the uniform distribution in the interval $[-1,1]$.

Question 1: What is the probability that $$\{f(x):x\in \mathbb R\} \subset [0,\infty)? $$i.e. that $f$ only takes non-negative values. Equivalently these $f$ can be written as $f=g^2+h^2$ for some $g,h \in \mathbb R[x]$. I think this probability is well-defined but I cannot prove nor disprove it. For $d=2$ the following easy argument shows that the probability, if it exists, is strictly positive: consider the set of polynomials $$ \{f=c_2x^2+c_1x+c_0:0\leq c_1<1/2<c_0,c_2\leq 1 \}$$ that has positive density. Each such polynomial has minimum value $$ \frac{-c_1^2+4c_0c_2}{4c_2}\geq \frac{-1/4+1}{4}=\frac{3}{16}>0.$$

Question 2: Now let us focus on the polynomials that only take non-negative values. As $f$ ranges among them, can we give a bound for the probability of the event $m_f>z,$ where $$m_f:=\inf\{f(x): x\in \mathbb R, f(x)>0\}?$$ I am only interested in rough bounds and only when $z\to+\infty$.