4. Find all real solutions to $x^{3}+(x+1)^{3}+(x+2)^{3}=(x+3)^{3}$.
Solution:
3
The equation simplifies to $3 x^{3}+9 x^{2}+15 x+9=x^{3}+9 x^{2}+27 x+27$, or equivalently, $2 x^{3}-12 x-18=2(x-3)\left(x^{2}+3 x+3\right)=0$. The discriminant of $x^{2}+3 x+3$ is $-3<0$, so the only real solution is $x=3$.