${{\left( a+b+c \right)}^{3}}-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}$$={{\left[ \left( a+b \right)+c \right]}^{3}}-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}$
$={{\left( a+b \right)}^{3}}+3{{\left( a+b \right)}^{2}}.c+3\left( a+b \right).c+{{c}^{3}}-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}$
$={{a}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}+{{b}^{3}}+3{{a}^{2}}c+6abc+3{{b}^{2}}c+{{c}^{3}}-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}$
$=3{{a}^{2}}b+3a{{b}^{2}}+3{{a}^{2}}c+3{{b}^{2}}c+6abc$
$=\left( 3{{a}^{2}}b+3{{a}^{2}}c+3abc \right)+\left( 3a{{b}^{2}}+3{{b}^{2}}a+3abc \right)$
$=3a\left( ab+ac+bc \right)+3b\left( ab+bc+ac \right)$$=3\left( ab+bc+ac \right).\left( a+b \right)$