how did you get from the 3rd to 4th line

*of the 1st qns

You don't need to do what I did under exams condition as the use of GC is expected.

To solve it analytically, you can either

(1) use partial fractions decomposition for repeated linear roots 2t^2/(t+1)^3 = A/(t+1) + B/(t+1)^2 + C/(t+1)^3 or

(2) create necessary (t+1)^n forms like what I did here such that the numerator and denominator can cancel. In this case you need to create them for n = 1 and 2 since the numerator goes up to quadratic (reverse engineering the required terms).

The end result for either method is the same.