Prove that $m^3\cdot 3^{k+3}-m^2\cdot 3^{2k+3}+m\cdot 3^{k+2}$ is divisible by $m\cdot3^{k+2}$

Question

Prove that $m^3\cdot 3^{k+3}-m^2\cdot 3^{2k+3}+m\cdot 3^{k+2}$ is divisible by $m\cdot3^{k+2}$ where $m$ is some integer. I have encountered this during a proof by induction problem.
I am not sure how to approach this problem. Could someone please help?

Answer 1

$$m^3\cdot 3^{k+3}-m^2\cdot 3^{2k+3}+m\cdot 3^{k+2}=\\ m\cdot 3^{k+2}\left(m^2\cdot 3-m\cdot 3^{k+1}+1\right)$$