<s>[INST] Find all real solutions for $x$ in $2(2^x−1)x^2+(2^{x^2}−2)x=2^{x+1}−2$. 
Find all real solutions for $x$ in $2(2^x−1)x^2+(2^{x^2}−2)x=2^{x+1}−2$.

I started off by dividing $2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2$ by $2$, and I got $(2^x-1)x^2 + (2^{x^2-1}-1)x = 2^x -1$. I tried dividing by $2^x-1$ on both sides which would give me a simple quadratic to solve for $x$, but I don't know how to simplify $\frac{(2^{x^2}-1)}{(2^x-1)}$. Am I missing a simple trick here, or am I on a completely wrong path to solving this?
 [/INST] Using Mathematica
eq = 2 (2^x - 1) x^2 + (2^(x^2) - 2) x == 2^(x + 1) - 2;

sol = Solve[eq, x, Reals]

(* {{x -> -1}, {x -> 0}, {x -> 1}} *)

Verifying the solutions
And @@ (eq /. sol)

(* True *)

Showing the solutions on a plot
Plot[{2 (2^x - 1) x^2 + (2^(x^2) - 2) x, 2^(x + 1) - 2},
 {x, -2, 2}, PlotLegends -> Placed["Expressions", {.22, .75}],
 Epilog -> {Red, AbsolutePointSize[5],
   Point[{x, 2^(x + 1) - 2} /. sol]}]


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