Which of the following sounds better (or more rigorously):
x grows with $N^2$ or
x grows with $O(N^2)$ or
x grows quadratically with $N$.
The three options have different meanings in mathematics:
You can rephrase the first two options as follows:
You mention rigor. Rigor is a property of proofs, whereas here there are no proofs. Your descriptions might be informal, but could help explain rigorous proofs.
The best two options are "$x$ grows quadratically in $N$" and $x = \Theta(N^2)$". Both are acceptable.