If 9x^2+24x+a is the square of a binomial, then the binomial has the form for some number , because . So, we compare to . Expanding gives
Equating the linear term of this to the linear term of 9x^2+24x+a, we have 6bx=24x, so b=4. Equating the constant term of 9x^2+6bx+b^2 to that of 9x^2+24x+a gives us a=b^2=16.