You've got an answer why do you say you're stuck.

Maybe you could simplify it a little but

dx/dt = {2t*(1-2t)+2t*t^2}/(1-2t)^2

dy/dt = -1/(1+t)^2 as you said

You did the good calculation dy/dx = (dy/dt)/(dx/dt)

Simplify as much as you can and that is your answer : (1/2)*{(2t-1)^2}/{t*(t^2-1)*(t+1)}