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.

Here's a slightly more motivated way to find a sum-of-squares representation.
We should want the first term to be the square of a quadratic with leading coefficient $1$, and to cancel out the $x^3$ term we need it to be $(x^2+2x+a)^2$ for some real $a$. This will leave us to show that
$$-2ax^2-(4a+4)x+(3-a^2)\geq 0;$$
the simplest case of this is to choose $a=-1$ (to remove the $x$ term entirely), which gives us that
$$x^4+4x^3+4x^2-4x+3=(x^2+2x-1)^2+2x^2+2>0.$$
