Next step to take in direct proof or a workaround around current dilemma?

Question

This is a problem from Discrete Math and Its Applications

I used a direct proof to do this proof. I understand the process/idea behind the direct proof, mainly (from https://courses.cs.washington.edu/courses/cse311/14au/slides/lecture07-filled.pdf)

Here is my work so far. Right now I am a mental roadblock. I did a substitution for n in the first equation and ended up with a proof that m - p is an even integer. Is there a workaround where I can get both positive m and p on one side(sum of the two) rather than the difference between the two?

Answer 1

Hint: You showed that $m-p$ is an even integer. Now note that $m+p=(m-p)+2p$.