Question 16327
The equation: {{{h = -16t^2 + 200t + 4}}} describes a parabola.  You need to find the location of the vertex of this parabola which will be a maximum since the parabola opens downwards (-16t^2).  The horizontal coordinate (t-component) of the vertex is given by: {{{-b/2a}}}. This corresponds to the time, t, at which the firework will reach its maximum height.

So, the maximum height will occur at: {{{-200/2(-16)}}} = {{{25/4}}}secs  = 6.25 seconds. This is when the firework should be detonated for it to explode at the greatest height.
To find this maximum height, substitute t = 6.25 into the original equation and solve for the height, h. It's easier to use the fractional form, 25/4

{{{h = -16(25/4)^2 + 200(25/4) + 4}}}
{{{h = -16(625/16) + 50(25) + 4}}}
{{{h = -625 + 1250 + 4}}}
{{{h = 629}}}feet.