<s>[INST] How can I solve this equation for real numbers? How can I solve this equation for real numbers?
$$(x+2)^4+x^4=82.$$
I tried $(x+2)^4+x^4-82=
2x^4 + 8 x^3 + 24 x^2 + 32 x - 66=0$
It is very difficult to continue.
 [/INST] Two solutions of $82=x^4+(x+2)^4$ are given by $x=1$ and $x=-3$. It follows that
$$ x^4+(x+2)^4-82 = (x-1)(x+3) q(x) $$
where $q(x)$ is a quadratic polynomial. Find it, solve $q(x)=0$ and you have the complete set of solutions ($q(x)=2\left[(x+1)^2+10\right]$, so the only real solutions are the trivial ones).
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