<s>[INST] How many ways can 4 numbers be arranged using +,-, ÷, and ×? How many ways is it possible to arrange 4 of the same number using addition, subtraction, multiplication and division. Not all of the signs must be used in a combination, for example: x+x+x+x is acceptable. Reverse combinations count as one (x+x+x-x and x-x+x+x).
 [/INST] I assume parentheses are not considered here, and the standard order of operations is used to evaluate the expression. The result is of the form: $$a_4x^4+a_3x^3+a_2x^2+a_1x+a_0+a_{-1}x^{-1}+a_{-2}x^{-2}$$
I also assume two of the expressions OP describes are equivalent if they are equal as rational expressions in $x$.
Think about the pair $(m,d)$ where $m$ counts how many times $\cdot$ was used, and $d$ counts how many times $\div$ was used. $(m,d)$ must be among the ten options: $$(0,0), (1,0), (2,0), (3,0), (0,1), (1,1), (2,1), (0,2), (1,2), (0,3)$$
$(m,d)=(3,0)$ implies the expression is $\color{blue}{x^4}$.
$(m,d)=(0,3)$ implies the expression is $\color{blue}{x^{-2}}$.
$(m,d)=(2,1)$ implies the expression is $\color{blue}{x^2}$.
$(m,d)=(1,2)$ implies the expression is $\color{blue}{1}$.
$(m,d)=(2,0)$ implies the expression is $\color{blue}{x^3+x}$, $\color{blue}{x^3-x}$, $\color{blue}{2x^2}$, $\color{blue}{0}$, or $\color{blue}{x-x^3}$.
$(m,d)=(0,2)$ implies the expression is $\color{blue}{x^{-1}+x}$, $\color{blue}{x^{-1}-x}$, $\color{blue}{2}$, $0$, or $\color{blue}{x-x^{-1}}$.
$(m,d)=(1,1)$ implies the expression is $\color{blue}{2x}$, $0$, $\color{blue}{x^2+1}$, $\color{blue}{x^2-1}$ or $\color{blue}{1-x^2}$.
$(m,d)=(0,0)$ implies the expression only uses $+$ and $-$, so it's of the form $nx$, and would be one of $\color{blue}{4x}$, $2x$, $0$, or $\color{blue}{-2x}$.
$(m,d)=(1,0)$ implies the expression must be one of  $\color{blue}{x^2+2x}$, $x^2$, $\color{blue}{x^2-2x}$, $\color{blue}{2x-x^2}$, $\color{blue}{-x^2}$ or $\color{blue}{-2x-x^2}$.
$(m,d)=(0,1)$ implies the expression must be one of  $\color{blue}{1+2x}$, $1$, $\color{blue}{1-2x}$, $\color{blue}{2x-1}$, $\color{blue}{-1}$ or $\color{blue}{-2x-1}$.
I count $29$ unique expressions (in blue). Of course this approach may come with human error on my part. Comments with corrections are welcome.
</s>