delta-epsilon proof, but it's not the proof that's stumped me!

Hi all, first post. I'm starting university next fall as a physics/math double major and I am taking Calculus I this summer so that I can take Physics I in the fall.

Here's the problem, we've been doing delta-epsilon proofs of limits. The class just started last thursday. I know how to do these proofs generally, and went through the homework. However, the professor asked if there were any math majors, and like an idiot I raised my hand, therefore I was "expected" to do an extra homework problem that he left out because it was "too hard."

Here it is:

Construct delta-epsilon proof.

I feel kind of embarrassed because I feel like I should be able to do it..

Prework:

And....

What would you do with

To get a | x - 4 | factor? As you can see, my problem is algebraic mainly...

Sorry if it's an easy one, I just don't know how to get my foot in the door!

Thanks!!