Determine all integer solutions of the equation:

$$
(x+1)^{3}+(x+2)^{3}+(x+3)^{3}+(x+4)^{3}=(x+5)^{3}
$$
For all integer $n$, $n^{3} \equiv n(\bmod 3)$, we should have: $(x+1)+(x+2)+(x+3)+(x+4) \equiv$ $x+5(\bmod 3)$, which is $4 x+10 \equiv x+5(\bmod 3)$, and this is impossible since $4 x \equiv x$ and $10 \not \equiv 5$ modulo 3.