Do the grouping $a^{5}+3a^{3}-8a^{2}-24=\left(a^{5}+3a^{3}\right)+\left(-8a^{2}-24\right)$, and factor out $a^{3}$ in the first and $-8$ in the second group.
Factor out common term $a^{2}+3$ by using distributive property.
$$\left(a^{2}+3\right)\left(a^{3}-8\right)$$
Consider $a^{3}-8$. Rewrite $a^{3}-8$ as $a^{3}-2^{3}$. The difference of cubes can be factored using the rule: $p^{3}-q^{3}=\left(p-q\right)\left(p^{2}+pq+q^{2}\right)$.
$$\left(a-2\right)\left(a^{2}+2a+4\right)$$
Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: $a^{2}+3,a^{2}+2a+4$.
