How to prove that $ P(f)\ast \delta=0 $ (P is the d'Alembertian)

Question

I am reading Distribution Theory by Friedlander and joshi (specifically chapter 7- coordinate transformations and pullbacks).

I find it hard to prove the next claim:

Show that if $P(\partial)= \partial_4^2-\partial_1^2-\partial_2^2-\partial_3^2$ is the definition of the D'alembertian then $ P(f)\ast \delta=0 $ where f is $C^\infty(X)$ and
X is an open subset of $\mathbb{R}$

Any clue?