Not sure what math class you're in, so I'll do it the "algebra" way as opposed to the calculus way.
Your formula is a quadratic, so its graph is a parabola that opens downward (since the leading coefficient, -16, is negative). So its vertex will be its highest point - remember that the vertex is the point where, in this case, the graph stops increasing and starts decreasing. So if we can find the vertex, we're done.
The x-coordinate for the vertex for a quadratic in the form is always given by:
In your case, a = -16, b = 50, so the x-coordinate of the vertex, which is represented by t, the time, in your equation, is (after some reducing). We need to know how HIGH the projectile gets - we now know that it reaches its highest point after seconds. To find its height, plug that time into the equation, giving:
I got about 44 seconds.