The question should state that the time is represented by x, not t.
a) The smallest value of (x-13)² is zero, when x=13, making the value in brackets maximum. So x is the time it reaches its max height of 12×169=2028 feet. Therefore the maximum height is reached in 13 seconds.
b) When H(x)=0, x(x-26)=0 and x=0 (start) or 26 seconds. So it takes 26 seconds to fall to the ground.