2016/011) Simplify $a^2(b+c)+b^2(c+a)+c^2(a+b)$ , if a,b,c are in Arithmetic Progression

$a^2(b+c)+b^2(c+a)+c^2(a+b)$
$= a^2(b+c+a) + b^2(c+a+b) + c^2(a+b+c) - (a^3+b^3+c^3)$
$= (a+b+c) (a^2+b^2+c^2) - ((a+b+c)(a^2+b^2 + c^2-ab-bc-ca) + 3abc)$
(because $a^3+b^3+c^3 - 3abc = (a+b+c) ( a^2+ b^2+ c^2 - ab- bc-ca))$
$= (a+b+c)( ab + bc + ca) - 3abc$
a b c are in ap so $a+c = 2b$ so $a+b+c = 3b$
now $ab+bc+ ca = b(a+c) + ca = 2b^2 + ca$
so sum = $3b(2b^2+ ca) - 3abc = 6b^3$