Hey kingsolomonsgrave.
You should try doing a quick expansion of the product. You will find that you can group the terms where both summands have the same index (i = j) and you will have a different expression where i != j.
As an example consider n = 2. You have
sigma (UiXi) * sigma(Ui) [1 to 2] = (U1X1 + U2X2)(U1 + U2)
= [U1^2X1 + U2^2X2] + [U1U2X1 + U1U2X2]
= sigma [1 to 2] (Ui^2*Xi) + sigma [i != j] [1 to 2] [1 to n] Ui*Uj*Xj
Use that pattern to show it for the general case of n > 1.