How do you factor 36m^3 - 6m^2 p^2 - 6mp + p^3?

Grouping works well here. Notice that the ratio of the first coefficient to the second is $- 6$ and the ratio of the third coefficient to the fourth is also $- 6$. So we find:
$36 {m}^{3} - 6 {m}^{2} {p}^{2} - 6 m p + {p}^{3}$
$= \left(36 {m}^{3} - 6 {m}^{2} {p}^{2}\right) - \left(6 m p - {p}^{3}\right)$
$= 6 {m}^{2} \left(6 m - {p}^{2}\right) - p \left(6 m - {p}^{2}\right)$
$= \left(6 {m}^{2} - p\right) \left(6 m - {p}^{2}\right)$