(8x^2y^2+4xy^2-12y^2) ÷ 4xy^2

I received this problem. I was told to divide the rational expression. The problem I am having is I feel like something is missing. I only understand to divide rational expressions we take the first rational expression and multiply it by the reciprocal of the 2nd rational expression. To me, this doesn't look right, it looks like maybe simplify? Or how would you divide this?
Thanks.. Steph

first term:

8x^2y^2/4xy^2=2x
then do the second, and third term.

for the third term, note
-12y^2/4xy^2=-3/x

Ok, this makes sense to me. I wrote out your explanation on paper and did the problem on paper. So, the answer is 2x-3x?

p.s. thanks so much for responding, really appreciate the help.

( 8 x ^ 2 y ^ 2 + 4 x y ^ 2 - 12 y ^ 2 ) / ( 4 x y ^ 2 ) =

4 y ^ 2 ( 2 x ^ 2 + x - 3 ) / ( 4 x y ^ 2 ) =

( 2 x ^ 2 + x - 3 ) / x

Thanks Bosnian. The way you worked it is actually what I had done before I posted the question, but I was unsure.

To divide the rational expression (8x^2y^2 + 4xy^2 - 12y^2) ÷ 4xy^2, you are correct that we can simplify first before dividing.

Let's break it down step by step:

Step 1: Simplify the numerator:
The numerator is 8x^2y^2 + 4xy^2 - 12y^2.
We notice that all terms have a common factor of 4y^2. We can factor it out:
8x^2y^2 + 4xy^2 - 12y^2 = 4y^2(2x^2 + xy - 3)

Step 2: Divide the simplified numerator by the denominator:
Now that the numerator is simplified, we can divide it by 4xy^2:
(4y^2(2x^2 + xy - 3)) ÷ (4xy^2)

Since the numerator and denominator have a common factor of 4, we can cancel it out:
= (y^2(2x^2 + xy - 3)) ÷ (xy^2)

Therefore, the simplified expression is y^2(2x^2 + xy - 3) ÷ (xy^2).

So, to answer your question, you were right about simplifying before dividing. In this case, we simplified the numerator by factoring out the common factor of 4y^2, and then we divided the simplified numerator by the denominator.