How to get from $(Ae^t.t)/(t+1)^2$ to $Ae^t/1+t - Ae^t/(1+t)^2$?

Question

I've been told that $(Ae^t.t)/(t+1)^2$ $=$ $Ae^t/1+t - Ae^t/(1+t)^2$ but im not sure how. I thought of using partial fractions but im not quite sure which case this would be.

Would appreciate the help.

Answer 1

$$\frac{Ae^t}{1+t}- \frac{Ae^t}{(1+t)^2}$$ $$=\frac{Ae^t}{1+t}\left(1-\frac{1}{1+t}\right)$$ $$=\frac{Ae^t}{1+t}\left(\frac{t}{1+t}\right)$$ $$=\frac{Ate^t}{(1+t)^2}$$

Answer 2

What is done is simply added and subtracted the $Ae^t/(1+t)^2$.