Let's determine $m$ such that

$$
x^{4}-5 x^{2}+4 x-m
$$

is divisible by $(2 x+1)$.
If we divide the expression $x^{4}-5 x^{2}+4 x-m$ by $(2 x+1)$, we obtain

$$
x^{4}-5 x^{2}+4 x-m=(2 x+1)\left(\frac{x^{3}}{2}-\frac{x^{2}}{4}-\frac{19 x}{8}+\frac{51}{16}\right)-\left(\frac{51}{16}+m\right)
$$

Therefore, our expression is divisible by $(2 x+1)$ if

$$
\frac{51}{16}+m=0
$$

which means if

$$
m=-\frac{51}{16}=-3 \frac{3}{16}
$$

(Borota Braniszláv, Szeged.)