Show that $f(a,b,c)=(a+b+c)^3+(a+b)^2+a$ is injective.

Question

For a function $f : \mathbb{N} \times \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ defined by

$f(a,b,c)=(a+b+c)^3+(a+b)^2+a$

I want to show that $f$ is injective.

How can I show this?

I started by assuming $(a,b,c) \neq (d,e,f)$ and $f(a,b,c)=f(d,e,f)$ and seeking a contradiction. However, I am uncertain about the subsequent steps.

Thank you.