Sorry to ask, but I couldn't find anywhere what a 'qual root' is.

About (C): if k=8, then $\displaystyle f(x) = x^2 + 8x + 24$

Now you must use the fact that $\displaystyle (a + b)^2 = a^2 + 2ab + b^2 \ (I)$, which is a particular case of Newton's Binomial. If you examine the expression for f(x), you can see that taking a = x and b = 4 and applying it to (I) seems promising. In fact:

$\displaystyle (x + 4)^2 = x^2 + 8x + 16$

Notice that it's not f(x) yet, but it's close. All you need to do now is adding 8:

$\displaystyle (x + 4)^2 + 8= x^2 + 8x + 16 + 8 = x^2 + 8x + 24 = f(x)$

And this answers your questions, a = 4 and b = 8. The name of this procedure is Square Completion. Check the link for further examples.

For (D), I'll leave it to you with a tip: does what you just did in (C) helps you to find the roots? Or inform you whether they exist in $\displaystyle \mathbb{R}$?

Regards,