I was trying to find the quote in the novel, but no luck as yet.
The clip you provide is where we discover that Raymond Boynton is depressed with life under his onerous stepmother, who tries to control the lives of all of her ‘children’.
The quote refers to Pandora’s Box, which stems from Greek mythology, where Zeus, the king of the gods, took vengeance on Prometheus (who stole fire from heaven) by presenting Pandora to Prometheus’s brother. Pandora opened a jar (box), left in his care and in doing so released a number of evil things into the world; sickness, death etc. However, one item was left behind as Pandora closed the lid, ‘hope’. So today if the expression ‘To Open Pandora’s Box’ is used it means you have opened a can of worms, you have done something that may get out of control, you have done something which you will not be able to control.
So, when Poirot talks to Raymond who is depressed and wants to live his own life and feels he has no control over it, Poirot suggests that while all things around him seem horrible and evil, he need not worry as there is always hope, which will prevail above all the evil things that abound.