To analyze the height of an object thrown upward, modeled by the polynomial function $h(t) = -16t^2 + 32t + 128$, we can find its zeros, the time it reaches a specific height, and its height at a specific time. First, to find the zeros which indicate when the object hits the ground, set $h(t) = 0$. Factoring the equation $-16t^2 + 32t + 128 = 0$ gives $-16(t^2 - 2t - 8) = 0$, which factors further to $-16(t - 4)(t + 2) = 0$. This yields $t = 4$ and $t = -2$. Discarding the negative time, the object hits the ground at 4 seconds. Second, to find when the object is at a specific height of 128 feet, set $h(t) = 128$. Subtracting 128 from both sides gives $-16t^2 + 32t = 0$. Factoring gives $-16t(t - 2) = 0$, yielding $t = 0$ and $t = 2$. The object is at 128 feet at 0 and 2 seconds. Third, to find the height at a specific time, such as $t = 1$ second, substitute 1 into the function. The result is $h(1) = -16(1)^2 + 32(1) + 128 = 144$. The object is at 144 feet after 1 second, which is its highest point.