Find one factor of the form $kp^{m}+n$, where $kp^{m}$ divides the monomial with the highest power $2p^{6}$ and $n$ divides the constant factor $24$. One such factor is $2p^{3}-3$. Factor the polynomial by dividing it by this factor.
$$\left(2p^{3}-3\right)\left(p^{3}-8\right)$$
Consider $p^{3}-8$. Rewrite $p^{3}-8$ as $p^{3}-2^{3}$. The difference of cubes can be factored using the rule: $a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right)$.
$$\left(p-2\right)\left(p^{2}+2p+4\right)$$
Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: $2p^{3}-3,p^{2}+2p+4$.
