Question:
Factorize: \[ a(a+b)^3-3a^2b(a+b) \]
Solution:
\[ a(a+b)^3-3a^2b(a+b) \]
\[ =a(a+b)\left[(a+b)^2-3ab\right] \]
\[ =a(a+b)\left[a^2+2ab+b^2-3ab\right] \]
\[ =a(a+b)(a^2-ab+b^2) \]
\[ \boxed{a(a+b)(a^2-ab+b^2)} \]
Factorize: \[ a(a+b)^3-3a^2b(a+b) \]
\[ a(a+b)^3-3a^2b(a+b) \]
\[ =a(a+b)\left[(a+b)^2-3ab\right] \]
\[ =a(a+b)\left[a^2+2ab+b^2-3ab\right] \]
\[ =a(a+b)(a^2-ab+b^2) \]
\[ \boxed{a(a+b)(a^2-ab+b^2)} \]