Hi, I have been working on understanding this question for a few days. I have figured out a bit but am at a loss with the end bit. So, here goes:

Write

(5/4x^2 - 25) - (2x+3) / ((4x^2+16x+15)/7)

as a single fraction in its simplest form.

The answer is :

(40-14x)/(4x^2-25)

I hope I have written it right.

In short, although it's taken me ages to get there, I have

looked at the right-hand side and dealt with division first:

(2x+3) / ((4x^2+16x+15)/7)

to become

7/(2x+5)

I was able to factorize the (4x^2+16x+15) to (2x+5)(2x+3) so that I could then divide the (2x+3) to get rid of it from top and bottom of the fraction to leave 7/(2x+5)

Then I had to deal with subtracting it:

(5/4x^2 - 25)- 7/(2x+5)

But I thought I had to have the denominators the same so I tried multiplying left side with (2x+5) and right side with this 4x^2 - 25 and ended up with

(5(2x+5)-7(4x^2-25))/((2x+5)(4x^2-25))

but its not right.

Any help would be great, thank you in advance.

I believe you intended to write 5/

**(**4x^2 - 25) - (2x+3) / ((4x^2+16x+15)/7).

If so, your work is correct through 5/

**(**4x^2 - 25) - 7/(2x+5).

What you need now is a common denominator. What you did would not make the LOWEST common denominator, requiring some extra simplification. (Also, you are probably expected to expand the numerator of your final answer - distribute and combine like terms - rather than leave it as you show it.)

Instead, first factor the first denominator, 4x^2 - 25. Then you can multiply the second fraction's numerator and denominator by just one linear factor. Then you will be able to add. (Presumably when you wrote about left and right sides, which are terms we use for sides of an equation, you meant the first and second fractions.)

I haven't checked for other errors.