You can do it with replacement patterns:
extractCommonFactor =
Sum[r_*s_, {sumIndex_, lmts__}] /; FreeQ[s, sumIndex] :>
s*Sum[r, {sumIndex, lmts}];
expandSquare = Sum[r_, {sumIndex_, lmts__}]^2 :>
Sum[r*(Sum[r, {sumIndex, lmts}] /. sumIndex -> sumIndex2),
{sumIndex, lmts}];
Sum[a[i] (n (n - 1) + 2 n), {i, 1, K}] - Sum[a[i] n, {i, 1, K}]^2 /.
extractCommonFactor /. expandSquare
Collect[%, n]