The term 'strongly encourage' can be broken down into four crossword solutions: EXHORT (6 letters), URGE (4 letters), EGGON (5 letters), and PUSHFOR (7 letters). EXHORT means to strongly URGE or advise someone to do something, while URGE is to strongly recommend or advise someone to do something. Eggon is an informal term meaning to strongly URGE or encourage someone to do something, while pushfor is to strongly advocate for something. All four of these terms are used to describe the action of strongly encouraging someone to do something.