Y= -16x2 + 32x - 10 this parabola has x-intercepts, representing the times when the dolphin's height above water is feet.

 This parabola has 0.3875 x-intercepts,
representing the times when the dolphin's height above water is 
0 feet.

The x-intercept is the point where the graph crosses the x-axis, which is when the value is y=0. 
The calculation for y=0 would be
Y = -16x2 + 32x - 10
0 = -16x2 + 32x - 10
-8x2 + 16x - 5=0

-b +-√(b^2-4ac)  /2a=
-16 +-√(16^2-4(-8)(-5))  /2(-8) =
-16+- √(256-160/ -16 =
-16+- √(96/ -16 =
-16+- 9.8/ -16 =
x1= 0.3875
x2= -1.6125