Find all solutions for \(x\) in \( 2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2 . \) Prove your answer
I checked and there are no mistypes. I have changed it to latex so you can look at it easier.
+30 Oh, the \(LaTeX\) version is much clearer.
Simply both sides first so you can isolate the variable.
Remember, isolate, isolate, isolate.
\(x=-1,0,1\)
\(x\) can be -1, 0, or 1.
+118667
\(2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2 \\ 2(2^x- 1) x^2 + (2^{x^2}-2)x = 2(2^{x} -1) \\ 2(2^x- 1) x^2- 2(2^{x} -1) =- (2^{x^2}-2)x \\ 2(2^x- 1) (x^2-1) =- (2^{x^2}-2)x \\\)
By inspection,
If both sides equal zero then they both must be the same.
If x=0 both sides are zero so that is a solution.
If x= +1 or -1 they are solutions too.
If x is bigger than 1 then the left side is positive and the right side is negative so no answers there
If x is less than -1 then the left side is negative and the right side is positive so no answer there either.
So could there be any more answers between -1 and +1?
I do not think so but i do not know how to prove it.
Maybe CuriousDude can show us both. 
