Suppose I have an even power polynomial $P(x)=x^4+a_1x^3+a_2x^2+a_1x+1$. How do find the interval for the values of $a_1$ and $a_2$ such that $P(x)$ has no solution in $\mathbb{R}$?
Additionally, how would I extend the method for finding the interval for any even power $P(x)$ with arbitrary coefficients?
I've only tried answering graphically, I have no idea to solve this analytically.
Clearly $x=0$ is not a solution, dividing by $x^2$ we get $$x^2+\frac{1}{x^2}+a_1\left(x+\frac{1}{x}\right)+a_2=0$$ now we substitute $$x+\frac{1}{x}=t$$ then we get $$x^2+\frac{1}{x^2}=t^2-2$$ and you will get a quadratic equation in $t$ to solve. $$t^2+a_1t+a_2-2=0$$ Can you proceed?