Find number of roots of the equation $e^x(x^4 + 4x^3 + 12x^2 + 24x + 24) + 1 = 0$

Question

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.

Answer 1

Just observe that

\begin{align} P(x) &=(x^4+4x^3+4x^2)+(8x^2+24x+18)+6\\ &=(x^2+2x)^2+2(2x+3)^2+6>0 \end{align}

Therefore $$e^xP(x)+1>0$$ for all $x \in \mathbb R$.