I am trying to figure out the concept behind inverse operators, but I am getting stuck. My book gives the definition as:

where is the series expansion of the inverse operator 1/P(D) obtained by ordinary division.

Okay - that sounds great, but what are they dividing by?

Here is the example in the book:

This is where I get lost. How did they go from step 1 to step 2?

Any help is greatly appreciated!