<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] Something awry should have jumped out at you when you wrote:
$$\color{blue}{\bf(a+b)^3} = (a+b)(a^2-ab+b^2) = \color{blue}{\bf a^3 + b^3}\;\;?$$
Just as $(a +b)^2 \neq a^2 + b^2$, it's also true that $(a+b)^3 \neq a^3 + b^3$.
So what went wrong? The difference of cubes does not apply here!
Rather, $(a+b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3.$ You can confirm that this is so by expanding the factors: $$(a + b)^3 = (a+b)(a+b)^2 = (a+b)(a^2 + 2ab + b^2) \cdots$$
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