Proving Big-Theta notation

Question

Hello I've tried my best to understand big-theta and now I get the main conception of the proofs for Big-Oh and Big-Omega but i couldn't find and example that is close to my excercise, because i cant do the proof for that one:

Prove, by exhibiting witnesses, that 4n^2 + 4n = Big-Theta(2n^2 + 32n)

I know that i have to prove it for Big-Oh and Big-Omega in order to prove Big-Theta, but i have no idea how to start. I mean the equation on the right side confuses me.

Answer 1

By the definition of big-theta, you need to show that there exist two constants, k1 and k2, such that for all sufficiently large values of n,

k1 * |2n^2 + 32n| <= |4n^2 + 4n| <= k2 * |2n^2 + 32n|

(Since your functions are all positive for positive n, you can drop the absolute values.) Just show that each inequality can be satisfied separately and you're done.

P.S. If this is homework, please tag it so.