Well, it seems this polynomial really knows how to spread the love (or the remainders, in this case). Let's see if we can keep up with its shenanigans!
We know that when the polynomial is divided by x-2, it leaves a remainder of 3. So, we can say that f(2) = 3. Similarly, when it's divided by x+2, it leaves a remainder of -5. This means that f(-2) = -5.
Now, we want to find the remainder when the polynomial is divided by x^2-4. Well, let's take a look at x^2-4 for a moment. Notice anything, uh, square about it? That's right, it's a difference of squares, (x+2)(x-2).
So, we have the remainders when f(x) is divided by both x-2 and x+2. And we also have the factorization of x^2-4 as (x+2)(x-2). It's starting to feel like a family reunion here!
Now, remembering our good old friend, the Remainder Theorem, we can use it to find the remainder. We'll substitute x = 2 and x = -2 into the polynomial, and that will give us the remainders.
Let's do the math before anyone says "clowning around":
f(2) = 3
f(-2) = -5
Now, let's find the difference between these remainders:
3 - (-5) = 8
So, the remainder when the polynomial is divided by x^2-4 is 8. Ta-da! We made it through the circus of polynomial remainders! I hope this answer has brought a smile to your face.