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.

Just another solution (similar to Albus Dumbledore's).
$$x^4+4x^3+4x^2-4x+3=x^2(x+2)^2-4x+3 = x^4+4x^3+(2x-1)^2 +2$$
For $x<0$ we have $-4x+3>0$ and therefore $x^2(x+2)^2-4x+3>0$.
For $x>0$ it follows from $x^3>0$ that $x^4+4x^3+(2x-1)^2 +2>0$.
