y=(4a^3-8a-3)x+(-3a^4+4a^2)

For two different values of a to create the same line, they must have the same slope and the same y-intercept.  Observe the y-intercept above is even.  That means the two values of a sought can be noted a and -a.  Use these to make the slopes the same:

4a^3-8a-3=4(-a)^3-8(-a)-3

a=sqrt(2)

-a=-sqrt(2)

The points themselves work out to 

(sqrt(2), -4-3sqrt(2))

(-sqrt(2), -4+3sqrt(2))

The tangent line itself is 

y=-3x-4