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.

if $x\ge 0$ by AM-GM $$x^4+3=x^4+1+1+1\ge 4x$$  and the inequality $$\color{blue}{x^4+3-4x}+4x^3+4x^2>0$$ is obvious as equality of am-gm and $x=0$ is simultaneously is not achieved.
If $x\le 0$ then replace $x$ by $-t $ and we have to prove $$t^4-4t^3+4t^2+4t+3>0$$which is true as $$t^4+4t^2\ge 4t^3$$ by AM-GM and as  $t\ge 0$ and the rest of terms are postive (note :again equality of both AM-GM and t=0 is not achieved)
