How did the industrialization of europe and the united states provide a motive for imperialism?

Imperialism ; the policy of extending the rule or authority of an empire or nation over foreign countries, or of acquiring and holding colonies and dependencies.

Industrialization ; to introduce industry into (an area) on a large scale

So if a country is mass producing products enabling its populace greater mobility in commerce, trade, war, education etc, it can be agreed that the imperial body will grow/expand.

The answer is installation.