How can polynomial identities be proven?

Polynomial identity has one side that consists of brackets. A way to prove identity is to get rid of brackets. Once you do that you just compare both sides of equal sign. if values on each side are the same than identity is true.

For example if identity is written by multiplication of factors just multiply them to get rid of brackets and compare it with side that doesn't have factors.

If a side has binomial on square, just square it (again you will get rid of brackets)  and compare it with other side.