We have to factorize the expression:
[tex]27m^3+125n^3[/tex]To do that we have to find the common factors we have in the terms of the expression. Sometimes they can be spotted without factorization, but if not, we can factorize each term and see the common terms that are present in all of them:
[tex]\begin{gathered} 27m^3=3^3m^3=(3m)^3 \\ 125=5^3n^3=(5n)^3 \end{gathered}[/tex]In this case, we don't have common terms but we have a sum of cubes. We can use the fact that a sum of cubes can be expressed as:
[tex]a^3+b^3=(a+b)\cdot(a^2-2ab+b^2)[/tex]In this case, we have a = 3m and b = 5n, so we can replace them in the equation above and solve as:
[tex]\begin{gathered} (3m)^3+(5n)^3=(3m+5n)\lbrack(3m)^2-2\cdot3m\cdot5n+(5n)^2\rbrack \\ (3m)^3+(5n)^3=(3m+5n)(9m^2-30mn+25n^2) \end{gathered}[/tex]Answer: the factorized form is (3m + 5n)(9m² - 30mn + 25n²)