<s>[INST] $x^{3}+ax^2+bx+c$ has all roots negative real numbers and a<3. Establish an inequality between only b and c 
A cubic equation $x^{3}+ax^2+bx+c$ has all negative real roots and $a, b, c\in R$ with $a<3.$
Prove that $b+c<4.$

My attempt :
Let the cubic be $f(x)$
Plotting graph we see that ,
$f(x\geq 0)>0$.
So we can see that a relation between $a, b, c$ can be established by putting $x=1$, so,
$1+a+b+c> 0$.
So $b+c>-4$.
Also by using Vieta's Formula we get $a, b, c > 0$.
Now I'm uncertain how to proceed, any help will be appreciated. This is a problem of a maths olympiad. Thanks.
 [/INST] We can write our polynomial in the form $(x+r_1)(x+r_2)(x+r_3)$, where $r_i$ are positive real numbers. Note that $b=r_1r_2+r_2r_3+r_3r_1$ and $c=r_1r_2r_3$.



$\color{Green}{\text{Maybe you will find}}$ Maclaurin's inequality $\color{Green}{\text{very interesting}}$. 
By Maclaurin's inequality, we know that $\sqrt[3]{S_3} \leq \frac{S_1}{3}$, 
or equivalently: 
$$c=r_1r_2r_3 = S_3 \leq (\frac{S_1}{3})^3 = (\frac{a}{3})^3 \leq (\frac{3}{3})^3=1.$$ 

Also again by use of Maclaurin's inequality, we know that $\sqrt[2]{\frac{S_2}{3}} \leq \frac{S_1}{3}$, 
or equivalently: 
$$b=r_1r_2+r_2r_3+r_3r_1 = S_2 \leq 3(\frac{S_1}{3})^2 = 3(\frac{a}{3})^2 \leq 3(\frac{3}{3})^3=3.$$ 

so $$b+c \leq 1+3=4.$$ 



$\color{Red}{\text{Comment}}$: Also, you can obtain all of the above inequalities 
$\color{Red}{\text{only}}$ by using AM–GM inequality. 
</s>