<s>[INST] Why cannot $(2x^2 + x)^2$ be simplified to $2x^4 + x^2$? So, I want to simplify an equation : $(2x^2 + x)^2$.
I thought this would simplify to $2x^4 + x^2$
But, if you input a value for $x$, the answers do not equal. For example, if you input $x = 3$, then: 
$$(2x^2+x)^2
= 21^2
= 441$$
AND:
$$2x^4 + x^2
= 2(82) + 9
= 173$$
Can anyone explain why this is the case?
 [/INST] The commutativity property states that:


*

*For all $\color{red}{a},\color{green}{b}$ we have $\color{red}{a}+\color{green}{b} = \color{green}{b}+\color{red}{a}$

*For all $\color{red}{a},\color{green}{b}$ we have $\color{red}{a}\times\color{green}{b} = \color{green}{b}\times\color{red}{a}$


distributivity property of multiplication over addition states that:


*

*For all $\color{red}{a},\color{green}{b},\color{blue}{c}$ you have: $(\color{red}{a}+\color{green}{b})\times \color{blue}{c} = \color{red}{a}\times \color{blue}{c} + \color{green}{b}\times \color{blue}{c}$


$(2x^2 + x)^2 = (\color{red}{2x^2}+\color{green}{x})\color{blue}{(2x^2+x)}$
For the moment, let us refer to the blue parenthesis as a single piece, and use the distributivity property above:
$=\color{red}{2x^2}\color{blue}{(2x^2+x)}+\color{green}{x}\color{blue}{(2x^2+x)}$
Now, using commutativity and distributivity, and reassigning colors, we see that this is:
$=(\color{red}{2x^2}+\color{green}{x})\color{blue}{2x^2} + (\color{red}{2x^2}+\color{green}{x})\color{blue}{x} = \color{red}{2x^2}\times \color{blue}{2x^2}+\color{green}{x}\times\color{blue}{2x^2}+\color{red}{2x^2}\times\color{blue}{x}+\color{green}{x}\times\color{blue}{x}$
$=4x^4+2x^3+2x^3+x^2=4x^4+4x^3+x^2$
</s>