Optimization Contest Problem: Prove $x^4+x^3-x^2-x+1>0$ for all real $x$.

In one of my previous blog posts(optimization-contest-problem), we want to prove that [MATH]\color{yellow}\bbox[5px,blue]{x^4+x^3-x^2-x+1}[/MATH] is always greater than zero for all real $x$, or more specifically, for $x\gt 1$.