summation - Series with Double Binomial Coefficients

How do I show the following?

$$ \sum_{x=0}^{n} x {N_1 \choose {n-x}} {N_2 \choose x} = N_2 {N_1 + N_2 - 1 \choose n-1} $$

I tried breaking down the left hand side into factorials and pulling out $N_2$, but that did not help. How does one deal with these summmations in general?

Answer

$$
\binom{N_2}{x} = \frac{N_2}{x}\binom{N_2-1}{x-1}
$$

With this, the sum gets transformed to

$$
\sum_{x=1}^n x\binom{N_1}{n-x}\binom{N_2}{x} = N_2\sum_{x=1}^n\binom{N_1}{n-x}\binom{N_2-1}{x-1}.
$$

The rest is easy with a combinatorial argument. Starting the index with $0$ or $1$ doesn't make a difference.