Prove that there not real roots with $P(x)=ax^3+bx^2+cx+d, $

Question

let $P(x)=ax^3+bx^2+cx+d,a,b,c,d\in R$, such that $$\min{\{d,b+d\}}>\max{\{|c|,|a+c|\}}$$

show that $P(x)=0$ have no real roots in $[-1,1]$

Answer 1

$P(-1)=-a+b-c+d$ and $P(1) = a+b+c+d$ so $P(-1)\times P(1)= -a^2+b^2-c^2+d^2-2ac+2bd=-(a+c)^2+(b+d)^2$ Since $Min\{d,b+d\}>Max${|c|,|a+c|}, we have $P(-1)\times P(1)>0$ and we have, granted by Bolzano's Theorem, that the number of roots of $P(x)=0$ in $[-1,1]$ is even, that means that $P(x)$ has $0$ or $2$ roots in $[-1,1]$.

Now uses the Cardano's formula to show that it cant have 3 distinct real roots, wich means that 2 roots are complex number and only one is real and this one cant be in $[-1,1]$