number of solutions in 0 to pi for sin3x=4sinxsin2xsin4x

sin 3x = 4 sinx sin2x sin4x

=> sin 3x = 2 sinx (2 sin 2x sin 4x)

=> sin 3x = 2 sinx {cos (2x-4x) - cos (2x+4x)}

=> sin 3x = 2 sinx (cos 2x - cos 6x)

=> sin 3x = 2 sinx cos 2x - 2 sin x cos 6x

=> sin 3x = {sin (x+2x) + sin (x-2x)} - {sin (x+6x) + sin (x-6x)}

=> sin 3x = sin 3x + sin (-x) - sin 7x - sin (-5x)

=> sin x + sin 7x + sin 5x = 0

Now, you can easily plot the curve above and see the number of points of intersection.

Curve is shown to you. (Approximate)

See the solutions between 0 to π.