$$f(x) = ax^2 + bx + c$$ < x

$$ax^2+x\left(b-1\right)+c\ <0$$

Here, $$x^2$$ is the significant term that influences the value of the expression. So, if the coefficient of this term is positive, irrespective of the other terms, the expression will be greater than x when the value of x goes higher.  

But if the value of a is negative, then this curve will be an inverted U-curve. So, $$a\le\ 0$$ is a correct condition.