Example 4 Let $a^{3}+b^{3}+3 a b=1$. Find the value of $a+b$.
[Analysis] Since the degree of the target algebraic expression is lower than that of the given algebraic expression, we need to reduce the degree of the given algebraic expression.
Let $a+b-k=0$. Suppose
$$
a^{3}+b^{3}+3 a b-1=(a+b-k) Q(x),
$$

where $Q(x)$ is a quadratic polynomial.
Let $a=0, b=k$.
From equation (1), we get
$$
k^{3}-1=0 \Rightarrow k=1 \Rightarrow a+b-1=0 \text {. }
$$

Therefore, $a^{3}+b^{3}+3 a b-1$
$$
\begin{array}{l}
=(a+b-1)\left(a^{2}-a b+a+b^{2}+b+1\right) \\
=0 .
\end{array}
$$

If $a^{2}-a b+a+b^{2}+b+1=0$, then
$$
\begin{array}{l}
a^{2}-a b+a+b^{2}+b+1 \\
=\frac{1}{2}\left((a-b)^{2}+(a+1)^{2}+(b+1)^{2}\right) \\
=0 .
\end{array}
$$

In this case, $a=b=-1$.
Therefore, $a+b=1$ or -2.