Hi

The inequality is true for

For instance for k=2 : for x=-3

There are several ways to show the inequality

You can study

f is differentiable and

The minimum of f is obtained for x=1 and f(1)=0 therefore

Second way : since f(1)=0 and f'(1)=0, f(x) can be factorized by (x-1)²

You can show that

and

Both factors are positive for x positive