Example 1 When $n$ is any real number and $k$ is a certain specific integer, the equation
$$
n(n+1)(n+2)(n+3)+1=\left(n^{2}+k n+1\right)^{2}
$$

holds. Then $k=$ $\qquad$ . [1]
(2010, Taiyuan Junior High School Mathematics Competition)
【Analysis】Since the left side of the given equation is a polynomial and the right side is in the form of a product, we only need to factorize the left side. The method of factorization is to use the overall idea and the complete square formula to handle it.
Solution Note that,
$$
\begin{array}{l}
n(n+1)(n+2)(n+3)+1 \\
=\left(n^{2}+3 n\right)\left(n^{2}+3 n+2\right)+1 \\
=\left(n^{2}+3 n\right)^{2}+2\left(n^{2}+3 n\right)+1 \\
=\left(n^{2}+3 n+1\right)^{2} .
\end{array}
$$

Therefore, $k=3$.