<s>[INST] Why does $(a + b)^3$ expand to $a^3 + 3ab^2 + 3a^2b + b^3$ 
Why does $(a + b)^3$ expand to $a^3 + 3ab^2 + 3a^2b + b^3$

Why does this work? I am confused as to why this happens. 
 [/INST] Let's break the algebra down, one step at a time. Trying to use colors to keep track of what comes from what.
You have that $$(a+b)^3 = \color{green}{(a+b)^2}\cdot (\color{#AA33FF}{a}+\color{#11AAFF}{b})$$
I assume you know $(a+b)^2 = a^2+2ab+b^2$, from which
$$
(a+b)^3 = \color{green}{(a^2+2ab+b^2)}\cdot (\color{#AA33FF}{a}+\color{#11AAFF}{b})
$$
Now, distributing (i.e., $c(a+b)=ca+cb$),
$$
(a+b)^3 = \color{green}{(a^2+2ab+b^2)}\cdot \color{#AA33FF}{a}+\color{green}{(a^2+2ab+b^2)}\cdot \color{#11AAFF}{b}
$$
Distributing again,
$$
(a+b)^3 = a^2\cdot \color{#AA33FF}{a}+2ab\cdot \color{#AA33FF}{a}+b^2\cdot \color{#AA33FF}{a} + a^2\cdot \color{#11AAFF}{b}+2ab\cdot \color{#11AAFF}{b}+b^2\cdot \color{#11AAFF}{b}
$$
that is
$$
(a+b)^3 = a^3+\color{red}{ 2a^2b}+\color{blue}{ a b^2} + \color{red} {a^2b}+\color{blue}{ 2ab^2}+b^3
$$
and regrouping the blue and red terms together,
$$
(a+b)^3 = a^3+\color{red}{3a^2b}+\color{blue}{3ab^2}+b^3
$$
giving the result.
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