Assume, to the contrary, that all roots of the polynomial $P(X)$ have nonegative real parts. We deduce that the real parts of the sums of the roots of its factors
$$a_{k} X^{4} + a_{k+1} X^{3} + a_{k+2} X^{2} + a_{k+3} X + a_{k+4}$$
for $k = 1, 2, 3, 4$, are nonegative. Therefore, by Vieta's relations, we have
$$\operatorname{Re} \left( -\frac{a_{k+1}}{a_{k}} \right) \geq 0$$
for $k = 1, 2, 3, 4$. Hence
$$-1 = \operatorname{Re} \prod_{k=1}^{4} \left( -\frac{a_{k+1}}{a_{k}} \right) = \prod_{k=1}^{4} \operatorname{Re} \left( -\frac{a_{k+1}}{a_{k}} \right) \geq 0,$$
which is a contradiction.

This proves that the polynomial $P(X)$ has a root with negative real part.