Keep in mind that x^2 - 4 is a "special product," whose factors are
x - 2 and x + 2. In this particular case this fact doesn't help much.
So, use either long division or synthetic division here:
Rewrite x^2 - 4 as x^2 - 0x - 4, whose coefficients are 1, 0 and -4.
Use -4 as the divisor in synth. div.; it comes from the divisor (x + 4).
Then the synth. div. looks like this:
-4 / 1 0 -4
-4 16
--------------------
1 -4 12
This result tells us that the quotient is 1x - 4 with a remainder of 12.
An interesting fact is that if you evaluate x^2 - 4 or x^2 - 0x - 4 at x = -4, the result will be 12 (same as the remainder shown above).