How to solve: $x^22^{x+1} + 2^{|x-3|+2} = x^22^{|x-3|+4} + 2^{x-1}$

Question

Any help would be appreciated. :)

I tried splitting the equation about $x=3$, but the terms $x^2$ and $2^x$ Together in the equation(s) are troubling me.
I don't know why I'm unable to apply the property $log_ax=\frac1{log_xa}$

Answer 1

HINT

I would say

$x^2 2^{x+1} + 2^{|x-3|+2} = x^2 2^{|x-3|+4} + 2^{x-1}$ |*2

$4x^2 2^{x} + 2^{|x-3|+3} = 4x^2 2^{|x-3|+3} + 2^{x}$

$4x^2 (2^{x} -2^{|x-3|+3})-(2^{x} -2^{|x-3|+3}) = 0$

$(4x^2 -1) (2^{x} -2^{|x-3|+3}) = 0$