By definition of tangent,
tan(x) = sin(x) / cos(x)
so if tan(x) < 0, and we're given cos(x) = -1/4 < 0, then it follows that sin(x) > 0.
Recall the Pythagorean identity:
cos²(x) + sin²(x) = 1 → sin(x) = + √(1 - cos²(x))
Then
sin(x) = √(1 - (-1/4)²) = √(15/16) = √(15)/4
Recall the double angle identity:
sin(2x) = 2 sin(x) cos(x)
Then
sin(2x) = 2 • √(15)/4 • (-1/4) = -2√(15)/16 = -√(15)/8
