I understand the difference between two squares but I don't understand this.
$\displaystyle
a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)
$
I think you mean $\displaystyle \displaystyle a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$
Start with $\displaystyle \displaystyle (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$ expand it out, what do you get?
Can you guess the following?
$\displaystyle \displaystyle a^3 - b^3 = \dots $
$\displaystyle \displaystyle a^4 - b^4 = \dots$
Expand $\displaystyle \displaystyle (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$ and simplify. Does it equal $\displaystyle \displaystyle a^5 - b^5$?
Can you guess the factorisations of $\displaystyle \displaystyle a^3 - b^3$ and $\displaystyle \displaystyle a^4 - b^4$? Can you see a pattern?