Derivation (1)

${\displaystyle (a+b)^{2}}$
${\displaystyle =(a+b)(a+b)}$
${\displaystyle =a(a+b)+b(a+b)}$
${\displaystyle =a^{2}+ab+ab+b^{2}}$
${\displaystyle =a^{2}+2ab+b^{2}}$

Derivation (2)

${\displaystyle (a-b)^{2}}$
${\displaystyle =(a-b)(a-b)}$
${\displaystyle =a(a-b)-b(a-b)}$
${\displaystyle =a^{2}-ab-ab+b^{2}}$
${\displaystyle =a^{2}-2ab+b^{2}}$

Derivation (3)

${\displaystyle a^{2}-b^{2}}$
${\displaystyle =a^{2}+ab-ab-b^{2}}$
${\displaystyle =a(a+b)-b(a+b)}$
${\displaystyle =(a+b)(a-b)}$

Derivation (4)

${\displaystyle (a+b)^{3}}$
${\displaystyle =(a+b)\times (a+b)^{2}}$
${\displaystyle =(a+b)\left(a^{2}+2ab+b^{2}\right)}$
${\displaystyle =a\left(a^{2}+2ab+b^{2}\right)+b\left(a^{2}+2ab+b^{2}\right)}$
${\displaystyle =a^{3}+2a^{2}b+ab^{2}+a^{2}b+2ab^{2}+b^{3}}$
${\displaystyle =a^{3}+3a^{2}b+3ab^{2}+b^{3}}$

Derivation (5)

${\displaystyle (a^{3}-b^{3})}$
${\displaystyle =(a-b)\times (a-b)^{2}}$
${\displaystyle =(a-b)\left(a^{2}-2ab+b^{2}\right)}$
${\displaystyle =a\left(a^{2}-2ab+b^{2}\right)-b\left(a^{2}-2ab+b^{2}\right)}$
${\displaystyle =a^{3}-2a^{2}b+ab^{2}-a^{2}b+2ab^{2}-b^{3}}$
${\displaystyle =a^{3}-3a^{2}b+3ab^{2}-b^{3}}$

Derivation (6)

Derivation (7)

Derivation (8)

Derivation (9)