Find a solution to the equation sin(4x) = 2 – sin 4x for 0 ≤ x ≤ π.?

That's only possible if sin 4x = 1, right?   If you don't see that, use algebra:

    sin 4x = 2 - sin 4x

    2 sin 4x = 2                  . . . add sin 2x to both sides

     sin 4x = 1                    . . . divide by 2

    4x = π/2 + 2πn            . . . all of the solutions to sin 4x = 1

    x = π/8 + πn/2             . . . divide by 4

So π/2 and 5π/8 are the values of x that are in [0, π].  The choices only include one of those, but don't forget that you can get multiple solutions to equations like this.

It looks like y = 2 - y

So then 2y = 2, which means that y = 1

Going back to the original, we get sin(4x) = 1.

You should be able to solve it now.