Example 16 Solve the equation
$$
(2 x+1)^{7}+x^{7}+3 x+1=0 .
$$
Solution: The original equation can be transformed into
$$
(2 x+1)^{7}+(2 x+1)=-\left(x^{7}+x\right) \text {. }
$$

Construct the function $f(x)=x^{7}+x$, the original equation can be rewritten as
$$
f(2 x+1)=-f(x) \text {. }
$$

Since $f(x)=x^{7}+x$ is an odd function, we have
$$
f(2 x+1)=f(-x) \text {. }
$$

Also, $f(x)=x^{7}+x$ is monotonically increasing over its domain $R$. By the property of monotonic functions, we get
$$
\begin{array}{l}
2 x+1=-x . \\
\therefore \quad x=-\frac{1}{3} .
\end{array}
$$
(3) Construct a graph