Simplyify by removing factors of 1?

Factor both the numerator and denominator

9t(t+1) / 9t(4t + 9)

Divide both numerator and denominator by 9t
(t+1)/(4t+9) where t is not 0

You need the "t is not 0" comment, because if t is 0, the initial fraction is undefined (dividing by 0 is a bad idea:).

(9t^2 + 9t)/(36t^2 + 81t)
= 9(t^2 + t)/9(4t^2 + 9t)
= 9t(t + 1)/9t(4t + 9) (cancel out 9t)
= (t + 1)/(4t + 9)

divide numerator and denominator by 9t

(t+1) / (4t+9)