Listen, Mr. Poirot. Get this — and get it clearly. I’m hiring you to investigate a case of murder.
-- Carla aka Clara Lemarchant (Introduction)

Importance: In her meeting with Poirot, Carla lets the investigator know clearly that she is wanting him to investigate the charges against her mother, murder charges from 16 years prior.

I can remember my mother — a patchy remembrance, of course, but I remember quite well the sort of person she was. She didn’t tell lies — kind lies.
-- Carla aka Clara Lemarchant (Introduction)

Importance: Carla tells Poirot that even though she was young when her mother died, she did remember that her mother was not one to lie. When Poirot talks to Williams, she concurs that Caroline was not one to lie.

Any ethics he had would have applied to painting. He wouldn’t, I’m convinced, have painted a sloppy, bad picture — no matter what the inducement. But for the rest, he was a full-blooded man and he...
-- Fogg (Book 1, Chapter 2)