Case 1:-
let , $x > 0 $ so , $|x|=x$
So , given expression ,
$5\left | x \right |+x^{2}-2x$
= $5x+x^{2}-2x$
=$x^{2}+3x$
Now with any value of $x>0$ this expression $x^{2}+3x>0$ .
Case 2:-
let , $x < 0 $ so , $|x|=-x$
So , given expression ,
$5\left | x \right |+x^{2}-2x$
= $5(-x)+x^{2}-2x$
=$x^{2}-7x$
Now for all value of $x<0$ , $x^{2}$>0 and $(-7x)$ is also positive as $x<0$ .
So , $x^{2}-7x$ >0 for all $x<0$.
Case 3:-
Let $x=0$ so , $|x|=0$
So , given expression ,
$5\left | x \right |+x^{2}-2x$
= $5*0+0^{2}-2*0$
=$0$
So, Given expression $5\left | x \right |+x(x-2) \geq 0$ for all real $x$ .