As I stated earlier, your mathematical work is sound.

$\displaystyle

1 = (-5)(t)^{2} + (20)(t) + 1

$

You start off by subtracting 1 on both sides, to yield:

$\displaystyle

0 = (-5)(t)^{2} + (20)(t)

$

Then you factored out -5t, to yield:

$\displaystyle

0 = (-5)(t) \times (t - 4)

$

Then you solved for t and got:

$\displaystyle

t = 0 ,4

$

All that makes sense to me, what I don't get is the logic of setting the height to 1. For part (b), if you know the time of ball in the air then could you not calculate the height for half the total time?