For all real numbers $x$, $y$ and $z$ prove that: $$3(x^2-x+1)(y^2-y+1)(z^2-z+1)\geq x^2y^2z^2-5xyz+x^2y+x^2z+y^2x+y^2z+z^2x+z^2y$$ I tried $uvw$ and more. For example, this inequality is a quadratic inequality of $x$ and it's enough to prove it for $\Delta\ge0$, but it's nothing.
I found this problem here :artofproblemsolving.com/community/c6h140p779898
My motivation is the following.
There is the following easy similar inequality.
For all non-negative $x$, $y$ and $z$ prove that: $$3(x^2-x+1)(y^2-y+1)(z^2-z+1)\geq1+xyz+x^2y^2z^2$$