Let's determine $m$ such that the expression

$$
m x^{2}+(m-1) x+m-1
$$

is negative for all values of $x$.

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Determine $m$ so that the expression

$$
m x^{2}+(m-1) x+m-1
$$

is negative for all values of $x$.
For the given expression to be negative for all values of $x$, it is necessary that the coefficient of $x^{2}$ be negative and that the curve representing the expression remain below the abscissa for all values of $x$. The equation $y=0$ must not have real or equal roots; for the roots to be complex, it is necessary that the discriminant

$$
y_{1}=(m-1)^{2}-4 m(m-1)
$$

be negative. If we geometrically represent this $y_{1}$ function, we see that the curve corresponding to the function intersects the abscissa at points whose abscissas are +1 and $-\frac{1}{3}$. The discriminant is therefore negative if $m>1$ or if $m<-\frac{1}{3}$. But $m$ cannot be positive according to the first condition, so the given expression is negative for all values of $x$ if $m<-\frac{1}{3}$.

The problem was solved by: Détshy K., Erdős A., Freibauer E., Goldziher K., Schwartz E., Szabó I., Szabó K., Weisz

J.