What is f(x) = 8x2 + 4x written in vertex form?

First of all, you need to complete the square of \(x\)

and this could be done as follows:

f(x)=8[x^2+0.5x]
     =8[x^2 + 0.5x + (0.5/2)^2 - (0.5/2)^2]
     =8[(x^2 + 0.5x +1/16) - 1/16]
     =8[(x+0.25)^2 - 1/16]

And our parabola has the vertex form of this:
 
    

and according to the chart I attached, the vertex is (-0.25,-0.5)

Tell me if there is any difficulty faces you!