Defining positive to be upward:

F = ma = 8|v| - mg

v is always directed downward. (A resistance can't propel the object upward, after all.) So...

m*dv/dt = -8v - mg

dv/dt + (8/m)v = g with v(0) = -10.

The homogeneous equation is

dv_h/dt + (8/m)v_h = 0

So

v_h(t) = Ae^{-(8/m)t}

And the particular solution looks like it's

v_p(t) = B

Putting this into the differential equation:

dv_p/dt + (8/m)v_p = g

0 + (8/m)B = g

B = mg/8

Thus

v(t) = v_h(t) + v_p(t) = Ae^{-(8/m)t} + (mg/8)

Now, v(0) = -10, so

-10 = A + (mg/8)

A = -10 - (mg/8) = -(mg + 80)/8

Thus

v(t) = -(1/8)(mg + 80)e^{-(8/m)t} + (mg/8)

-Dan