The equation for a projectile's height versus time is . a tennis ball machine serves a ball vertically into the air from a height of two feet, with an initial speed of 120 feet per second. after how many seconds does the ball attain its maximum height?

The time it takes the ball to reach its maximum height only depends on
its initial vertical speed.  The height from which it's launched makes no
difference.

Since you're still reporting height in 'feet' and speed in 'feet per second',
we can assume that you haven't achieved space travel yet, and this whole
episode with the tennis ball is taking place on Earth.  So we know that the
acceleration of gravity is 32.2 ft/s², and the ball will lose 32.2 ft/s of its
upward speed every second.  Naturally, it'll stop rising when its vertical
speed has shriveled to zero.

That happens in  (120/32.2) = 3.73 seconds after the launch.