Existence of negative root for a quartic polynomial

Question

Be $P(X)=X^4+aX^3+bX^2+cX+d$ a quartic polynomial with real coefficients such that $a<0$ and $d>0$.

Are there sufficient conditions on the coefficients so that P has at least one negative root ?

Answer 1

If $P(x) < 0$ for some $x < 0$, there are at least two negative roots: one in $(-\infty, x)$ and one in $(x, 0)$. For example, with $x = -1$ the sufficient condition is $1 - a + b - c + d < 0$.

Answer 2

This direction is directly looking at necessary conditions. If we demand a real root $-e$ where $e>0,$ taking $e,t,v >0,$ we find $$(x+e) \left( x^3 -(t+e)x^2 + u x + v \right) = x^4 - t x^3 + (u - e^2 - et)x^2 + (v+eu) +ve $$

Thus, $a = -t$ and so on. Note $d=ve$ is positive, as demanded.