Let $r$ be a root of $x^2 + 5x + 7 = 0.$  Compute
\[(r - 1)(r + 2)(r + 6)(r + 3).\]
Since $r$ is a root of $x^2 + 5x + 7 = 0,$ $r^2 + 5r + 7 = 0.$

We have that
\begin{align*}
(r - 1)(r + 2)(r + 6)(r + 3) &= (r - 1)(r + 6)(r + 2)(r + 3) \\
&= (r^2 + 5r - 6)(r^2 + 5r + 6) \\
&= (-13)(-1) = \boxed{13}.
\end{align*}