Let's determine $m$ such that

$$
2 x^{4}+4 a x^{3}-5 a^{2} x^{2}-3 a^{3} x+m a^{4}
$$

is divisible by $(x-a)$.
If we divide the given expression by $(x-a)$, then we obtain:

$$
\begin{gathered}
2 x^{4}+4 a x^{3}-5 a^{2} x^{2}-3 a^{3} x+m a^{4}= \\
=\left(2 x^{3}+6 a x^{2}+a^{2} x-2 a^{3}\right)(x-a)+m a^{4}-2 a^{4}
\end{gathered}
$$

thus, our expression is divisible by $(x-a)$ if

$$
m a^{4}-2 a^{4}=0, \quad \text { that is, if } m=2
$$

(János Szilvay, Eperjes.)