The function h(t) = –16t^2 + 96t + 6 represents an object projected into the air from a cannon. the maximum height reached by the object is 150 feet.
the possible answers are :
2 seconds
3 seconds
6 seconds
9 seconds

May I presume that the question is "How long does it take
the object to reach its maximum height ?" ?

If you've started pre-calculus, then you know that the derivative of  h(t)
is zero where  h(t)  is maximum.

The derivative is            h'(t) = -32 t  +  96 .

At the maximum ...        h'(t) = 0

                                       32 t = 96 sec

                                           t  =  3 sec .


If you haven't had any calculus yet, then you don't know how to
take a derivative, and you don't know what it's good for anyway.

In that case, the question GIVES you the maximum height.
Just write it in place of  h(t), then solve the quadratic equation
and find out what  't'  must be at that height.

                                       150 ft = -16 t²  +  96  t  +  6

Subtract 150ft from each side:    -16t²  +  96t  -  144  =  0 .

Before you attack that, you can divide each side by  -16,
making it a lot easier to handle:

                                                         t²  -  6t  +  9  =  0

I'm sure you can run with that equation now, and solve it.   
The solution is the time after launch when the object reaches 150 ft.
It's 3 seconds. 
(Funny how the two widely different methods lead to the same answer.)