Problem solutions should be reasonably self-contained and clear (they should be clear enough to one of your classmates); formulas and computations must be accompanied by explanations in ordinary formal English. Typical rules of writing style also apply to mathematical texts! I encourage you to reflect about effective mathematical writing.

A short reference: Su, “Guidelines for good mathematical writing”

The most celebrated book on this topic might be 

  • Norman Earl Steenrod, Paul R. Halmos, Menahem M. Schiffer and Jean R. Dieudonné, How to write mathematics, American Mathematical Society, 1973

which is a collection of essays authored by some of the finest mathematical writers of the XXth century (particularly Halmos and Dieudonné, which are archetypical representatives of the American and French schools, respectively). The book 

  • Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts, Mathematical Writing, The Mathematical Association of America, 1996

is also a classic. You may already know that Knuth created TeX, so you can guess that he also took writing quite seriously.  

If you prefer videos, there’s a delightful lecture by Serre on the topic

https://www.youtube.com/watch?v=tJZpdXWm4Gg

unfortunately only in very low resolution. 

Concerning proofs, you could read: https://math.berkeley.edu/~hutching/teach/proofs.pdf