Question

How do you divide `\frac{a^2+2ab+b^2}{ab^2-a^2b} \-: (a+b)`?

Answer 1

We can use the rule about division of rational expressions where you can change the division in a multiplication by flipping the second fraction.
In our case you have (remember that `(a+b)` can be written as a fraction of `(a+b)/1`):

`(a^2+2ab+b^2)/(ab^2-a^2b)-:(a+b)/1=(a^2+2ab+b^2)/(ab^2-a^2b)xx1/(a+b)=`

You can also rearrange the nominator and denominator of the first fraction as:

`a^2+2ab+b^2=(a+b)^2` and:
`ab^2-a^2b=ab(b-a)`

So, finally:

`(a^2+2ab+b^2)/(ab^2-a^2b)xx1/(a+b)=((a+b)^2)/(ab(b-a))xx1/(a+b)=`
`=(a+b)/(ab(b-a))`