summation - Sum the first $n$ terms of the series $1 cdot 3 cdot 2^2 + 2 cdot 4 cdot 3^2 + 3 cdot 5 cdot 4^2 + cdots$

Sum the first $n$ terms of the series:

$$1 \cdot 3 \cdot 2^2 + 2 \cdot 4 \cdot 3^2 + 3 \cdot 5 \cdot 4^2 + \cdots.$$

This was asked under the heading using method of difference and the answer given was

$$S_n = \frac{1}{10}n(n+1)(n+2)(n+3)(2n+3).$$

First, I get

$$U_n=n(n+2)(n+1)^2.$$

Then I tried to make $U_n = V_n - V_{n-1}$ in order to get $S_n = V_n - V_0$. But I really don't know how can I figure this out.