<s>[INST] Given $P(x)=x^{4}-4x^{3}+12x^{2}-24x+24,$ then $P(x)=|P(x)|$ for all real $x$ $$P(x)=x^{4}-4x^{3}+12x^{2}-24x+24$$
I'd like to prove that $P(x)=|P(x)|$. I don't know where to begin. What would be the first step?
 [/INST] As the other answers pointed out, you need to prove that the polynomial is positive for all values of $x$. A very common way to do it is to express it as a sum of squares.
$$P(x)=x^4−4x^3+12x^2−24^x+24$$
The problems here are the two terms where X is at an odd power: $-4x^3$ and $-24x$
Trying to fit them into squares is almost automatic:
\begin{align}
P(x) & = x^4−4x^3+12x^2−24x+24 \\
     & = (x^4 - 4x^3 + 4x^2) + 8X^2 - 24X + 24 \\
     & = x^2(x-2)^2 + 6(x^2 - 4X + 4) + 2x^2 \\
     & = x^2(x-2)^2 + 6(x-2)^2 + 2x^2
\end{align}
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