How to simplify polynomial to get to particular factor?


If I have an expression like this

ex = -l + l t - 2 t^2 + T + (l - 2 t) y 

This can be written as

ex1 = -l + (l - 2 t) (t + y) + T 

However, if I try

FullSimplify[ex] 

I get

T + l (-1 + t + y) - 2 t (t + y) 

How can I get the form of ex1 without having to manually copy, paste and modify?