x=0 is an obvious solution. Also x=π and 2π because all integer multiples of π have a zero sine.
3sin(x)=sin(π-x), so if sin(4x)=sin(x), 4x=π-x, 5x=π, x=π/5. x=-π/5 will also be solution because sin(-x)=-sin(x) (sine is an odd function). But sin(-x)=sin(2π-x), therefore x=2π-π/5=9π/5 is another solution.
sin(x)=sin(x+2πn) where n is an integer. so 4x=x+2πn, 3x=2πn, x=2πn/3. When n=1, x=2π/3; when n=2, x=4π/3.
Also sin(4x)=sin(4x-2πn)=sin(π-x), 4x-2πn=π-x, 5x=π+2πn. When n=1, x=3π/5; when n=3, x=7π/5.
So we have all 9 solutions: x=0, π/5, 3π/5, 2π/3, π, 4π/3, 7π/5, 9π/5, 2π.