Let's determine $m$ such that the expression

$$
(4-m) x^{2}-3 x+4+m
$$

is positive for all values of $x$.

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Determine $m$ so that the expression

$$
(4-m) x^{2}-3 x+4+m
$$

is positive for all values of $x$.
The function $f$ of $x$ is positive for all values of $x$ if its minimum is positive, or if

$$
m-\frac{\sqrt{55}}{2}, \quad \text { or } \quad m<\frac{\sqrt{55}}{2}
$$

Since $\frac{\sqrt{55}}{2}<4$, the required condition is

$$
-\frac{\sqrt{55}}{2}<m<\frac{\sqrt{55}}{2}
$$