jmc

We can use the Euclidean Algorithm. $$\begin{array}{rl}& \text{gcd}(a^2+9a+24,a+4)\\ & =\text{gcd}(a^2+9a+24-(a+5)(a+4),a+4)\\ & =\text{gcd}(a^2+9a+24-(a^2+9a+20),a+4)\\ & =\text{gcd}(4,a+4).\end{array}$$ Since $4$ is a factor of $a$ and thus $a+4$, the greatest common divisor is $\boxed{4}$.