Example 6 Given real numbers $x, y$ satisfy $(3 x+y)^{5}+x^{5}$ $+4 x+y=0$. Find the value of $4 x+y$.
Given that it can be transformed as
$$
(3 x+y)^{5}+(3 x+y)=-\left(x^{5}+x\right) \text {. }
$$

Let $f(t)=t^{5}+t$ be a monotonic odd function. Equation (1) indicates that the two function values are equal, i.e.,
$$
f(3 x+y)=-f(x)=f(-x) .
$$

Therefore, the independent variables are equal, which gives
$$
3 x+y=-x,
$$

i.e., $4 x+y=0$.