edit: I read the question again.
here is how I would do it:
you know that A = 1/2(21-h)*h = 10.5h - 0.5h^2
then dA/dh = 10.5 - h
when dA/dh = 0, h = 10.5
when h = 10.5, b = 10.5, so A = 55.125 cm^2
So here is the question and the way I worked through it
The answer is 55.125 cm^2
I just feel as thought my method used to arrive at the answer is either wrong, or a misinterpretation of steps in order
It's really been bothering me, like there is an approach to the question, which is the correct procedure, and I haven't implemented it.
Take a look ladies & gents.
@OP: You have made heavy weather of finding the zeros of the area function. Since the original expression was factorised, it ought to be evident that the zeroes are h = 21 and h = 0 using the null factor law (you might know this idea under a different name). The maximum turning point will therefore lie halfway between - at x = 10.5 etc.
The question appears to be dealt with satisfactorily so for various reasons I'm closing this thread.