Phân tích thành nhân tử a(a+2b)^3 -b(2a+b)^3

* Hướng dẫn giải

$$\begin{gathered} a(a+2b{)}^3-b(2a+b{)}^3 \\ =a{\left[\left(a+b\right)+b\right]}^3-b{\left[a+\left(a+b\right)\right]}^3 \end{gathered}$$

$=a\left[{\left(a+b\right)}^3+3b{\left(a+b\right)}^2+3b^2\left(a+b\right)+b^3\right]-b\left[a^3+3a^2\left(a+b\right)+3a{\left(a+b\right)}^2+{\left(a+b\right)}^3\right]$

$=a{\left(a+b\right)}^3+3ab{\left(a+b\right)}^2+3a{b}^2\left(a+b\right)+a{b}^3-b{a}^3-3b{a}^2\left(a+b\right)-3ab{\left(a+b\right)}^2-b{\left(a+b\right)}^3$

$=\left(a-b\right){\left(a+b\right)}^3+3ab\left(a+b\right)\left(b-a\right)+ab\left(b-a\right)\left(b+a\right)$

$=\left(a-b\right)\left(a+b\right)\left[{\left(a+b\right)}^2-3ab-ab\right]$

$=\left(a-b\right)\left(a+b\right)\left(a^2-2ab+b^2\right)$

$=\left(a-b\right)\left(a+b\right){\left(a-b\right)}^2$

$=\left(a+b\right){\left(a-b\right)}^3$