Find number of roots of the equation $e^x(x^4 + 4x^3 + 12x^2 + 24x + 24) + 1 = 0$
Using Descartes rule, number of positive roots is zero and there can be a maximum of 4 negative roots.
Also, for the function $P(x)=x^4 + 4x^3 + 12x^2 + 24x + 24$, the double derivative $P''(x)>0$. So the function can have either 2 negative roots or no root at all.
But I still lack information required to prove that the function has no real roots.