<s>[INST] How does $(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$? Good afternoon my wonderful friends!
Whenever I do this equation I set it up using the difference of two cubes, which is as follows:
$(a+b)^3 = (a+b)(a^2-ab+b^2) = a^3 + b^3$
Whenever I try to use this formula  I always get:
$(x+h) x^2  - xh + h^2$

Simplify:

$x^3 - x^2h + h^2x + x^2h - xh^2 + h^3$
Simplify:

x^3 + h^3$
I don't understand were the three's come from in the final answer: $(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$
 [/INST] I don't know where you get "$a^2-ab+b^2$" from. Just use distributivity to carry out the multiplications by hand:
\begin{align}
(a+b)^2 &= (a+b)(a+b) = a(a+b)+b(a+b) \\&= a^2+ab+ba+b^2 = a^2+2ab+b^2
\end{align}
Thus,
\begin{align}
(a+b)^3 &= (a+b)(a+b)^2 = (a+b)(a^2+2ab+b^2) = a(a^2+2ab+b^2)+b(a^2+2ab+b^2) \\&= a^3+2a^2 b+ab^2 + ba^2 + 2ab^2+b^3 = a^3 + 3a^2b + 3ab^2 + b^3.
\end{align}
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