$(a + b)(a - b) = a^2 - b^2$

Theorem: $(a + b)(a - b) = a^2 - b^2$

Prerequisites:
Distributivity property of multiplication

Proof:
Applying distributivity property of multiplication repeatedly on LHS:

$\begin{align*} \qquad\quad LHS &= (a + b)(a - b)\\ &= (a + b)a - (a + b)b \\ &= (a^2 + ab) - (ab + b^2) \\ &= a^2 - b^2\\ &= RHS \end{align*}$

Hence the result


Recommended:
$(a + b)^2 = a^2 + b^2 + 2ab$
$(a + b)^3 = a^3 + 3a^2b + 3ab^2+b^3$