To understand how to prove something algebraically, we must first ask ourselves: "What is a proof?"
A proof is simply a logical, step-by-step guide to demonstrate that a given rule or statement is ALWAYS correct. You cannot, therefore, prove something simply by giving an example, as this does not make something ALWAYS correct; although you may have demonstrated that your example fits the rule, you have not shown that the rule is correct in every instance.
The best way to mathematically prove something is using algebra. Sometimes, a question may already give you the algebraic notation, for example:
"Prove that (2a - 1)^2 - (2b - 1)^2 = 4(a - b)(a + b - 1)"
Here, you would simply rearrange the left side to make it look like the right side of the equation, writing down each stage in your working, like so:
(2a - 1)^2 - (2b - 1)^2
Expanding the brackets gives:
4a^2 - 2a - 2a + 1 - 4b^2 + 2b + 2b - 1
Collecting like terms gives:
4a^2 - 4a - 4b^2 + 4b
Taking out a factor of 4 gives:
4(a^2 - b^2 - a + b)