I need help with this

OK, the correct answer is \(b^2 +4b - 2 + \dfrac{3}{4b-6}\). 

Note that when you subtract a multiple of 4b-6, you must change all of the terms. For 4b-6 to have a b^3, you should multiply 4b-6 by b^2 to get 4b^3 - 6b^2. You cannot directly change it to \(-4b^3 +4\) like you did in the image. 

Knowing this, you should be able to determine the answer. 

- PM

Good job, PM !!!!!

Here's how PM arrived at his answer.....

                b^2   + 4b    -   2

4b - 6   [  4b^3 +  10b^2 - 32b + 15 ]

               4b^3 -     6b^2

              ______________________

                            16b^2  - 32b

                            16b^2 -  24b

                            ________________

                                       -8b  + 15

                                       -8b +  12

                                       _________

                                                   3