<s>[INST] Show that if $a\neq b$ then $a^3+a\neq b^3+b$ Show that if $a\neq b$ then $a^3+a\neq b^3+b$
We assume that $a^3+a=b^3+b$ to show that $a=b$
$$\begin{align}
a^3+a=b^3+b &\iff a^3-b^3=b-a\\
&\iff(a-b)(a^2+ab+b^2)=b-a\\
&\iff a^2+ab+b^2=-1
\end{align}$$
Im stuck here !
 [/INST] I'm not a math major so I'm not sure about the formalism but does this work?
If a!=b,
Then a=b+d ; where d is a non zero integer.
Plug in to the expression (a^3+a)-(b^3+b) :-
3bd^2+3db^2+d^3.
This has only one real root which is d=0, which is a contradiction, so the two expressions cannot be equal. 
Too informal? Correct or incorrect, math majors?
</s>