Notice first that $2014 = 2 \times 19 \times 53$. Applying Fermat's little theorem for the prime numbers $2, 19, 53$ we obtain:
$$\begin{gathered} 53 n^{55} - 57 n^{53} + 4 n \equiv n - n + 0 \equiv 0 \quad \bmod 2 \\ 53 n^{55} - 57 n^{53} + 4 n \equiv 15 n^{17} (n^{19})^{2} - 0 + 4 n \equiv 15 n^{19} + 4 n \equiv 19 n \equiv 0 \quad \bmod 19 \\ \text{and } 53 n^{55} - 57 n^{53} + 4 n \equiv 0 - 4 n + 4 n \equiv 0 \quad \bmod 53 \end{gathered}$$
for all integer $n$. Therefore $2014$ divides $53 n^{55} - 57 n^{53} + 4 n$ for all integer $n$.