Find one factor of the form $ka^{m}+n$, where $ka^{m}$ divides the monomial with the highest power $2a^{6}$ and $n$ divides the constant factor $-24$. One such factor is $2a^{3}+3$. Factor the polynomial by dividing it by this factor.
$$\left(2a^{3}+3\right)\left(a^{3}-8\right)$$
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: $2a^{3}+3,a^{2}+2a+4$.
