Use of $E( X(X-1) )$ to calculate variance of random variable

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In an exercise I had to calculate the variance of the Poisson-distribution.
In the solution, $E( X(X-1) )$ was used, which for me fell out of thin air:

Solution

I can see how this works out, but I'd like to know some context.

Is this a common technique?
Does it have a name?
Are there similar "tricks" to calculate the variance?