Chứng minh hằng đẳng thức: (a+b+c)^3 -a^3 -b^3 -c^3 = 3(a+b)

* Hướng dẫn giải

$$\begin{gathered} {\left(a+b+c\right)}^3-a^3-b^3-c^3=\left[{\left(a+b+c\right)}^3-a^3\right]-\left[b^3+c^3\right] \\ \begin{array}{l}=\left(b+c\right)\left[{\left(a+b+c\right)}^2+a\left(a+b+c\right)+a^2\right]-\left(b+c\right)\left(b^2-bc+c^2\right)\\ =\left(b+c\right)\left(3a^2+3ab+3ac+3bc\right)\\ =3\left(a+b\right)\left(b+c\right)\left(c+a\right)\end{array} \end{gathered}$$