<s>[INST] Prove the polynomial $x^4+4 x^3+4 x^2-4 x+3$ is positive Given the following polynomial
$$
x^4+4 x^3+4 x^2-4 x+3
$$
I know it is positive, because I looked at the graphics

and I found with the help of Mathematica that the following form
$$
(x + a)^2 (x + b)^2 + c^2(x + d)^2 + e^2
$$
can represent the polynomial with the following values for the constants
$$
\left(x-\frac{1}{2}\right)^2
   \left(x+\frac{5}{2}\right)^2+\frac{5}{2} \left(x+\frac{1}{5}\right)^2+\frac{107}{80}
$$
I suppose there are simpler ways to prove that the polynomial is positive, perhaps by using some inequalities.
Please, advice.
 [/INST] Let the polynomial be denoted by $f(x)$.
$f'(x)=4(x^3+3x^2+2x-1)$ has only one real root $\approx 0.3247$. At that point $f''(x)=4(3x^2+6x+2)>0$. Thus $f(x)$ has minimum value at $\approx 0.3247$ and $f(0.3247)\approx 2.271$.
$\therefore f(x)>0 \;\forall x\in \mathbb R$.
</s>