To make this equation have infinitely many solutions, we want to end up with the equation x=x. With this equation, no matter what number we plug in for x, it will be a solution since it will equal itself.
-(x-8)+4x=2()+x
-x+8+4x=2()+x
-2x+8+4x=2()
2x+8=2()
x+4=()
Therefore if we plug x+4 into the equation we will find our equation with infinitely many solutions.
-(x-8)+4x=2(x+4)+x
We can check our solution:
-(x-8)+4x=2(x+4)+x
-x+8+4x=2x+8+x
3x+8=3x+8
Since the right and left side of the equation our equal, we know our solution works.