Example 6. Solve the equation:
$$
(x+3)^{4}+(x+1)^{4}=82
$$
Let $y=x+2$,
the original equation becomes $(y+1)^{4}+(y-1)^{4}=82$.
After simplification, we get $y^{4}+6 y^{2}-40=0$.
Solving for $y$ yields $y= \pm 2$.
From $x+2= \pm 2$, we obtain the solutions of the original equation:
$$
x_{1}=0, x_{2}=4 .
$$