Turn [2x+3 / 5x+7] into [(2 + 3/x) / (5 + 7/x)]?

2x + 3
---------------
5x + 7

Divide top and bottom by x:-

2 + 3/x
--------------
5 + 7/x

You can divide both top and bottom of the fraction by x so:

[ ((2x+3)/x) / ((5x+7)/x) ]

looking at the top: (2x+3)/x can be split up into 2x/x + 3/x which is 2+3/x

Looking at the bottom: (5x+7)/x can be split up into 5x/x + 7/x which is 5 + 7/x

Leaving us with [ (2+3/x) / (5+7/x) ]

divide both the top and the bottom by x
ie top:
(2x-3)/x
=(2x/x)-(3/x)
=(2)-(3/x)

bottom:
(5x+7)/x
=(5)+(7/x)

so by dividing both bottom and top by x you are multiplying the entire thing by x/x
x/x=1
therefore they are equal

Divide ever term by x
((2x/x)+(3/x))/((5x/x)+(7/x)) will give you
(2 + 3/x)/ ( 5 + 7/x)

(2x + 3)/(5x + 7)
= [(2x + 3)/x]/[(5x + 7)/x]
= [2x/x + 3/x]/[5x/x + 7/x]
= [2 + 3/x]/[5 + 7/x]

To turn (2x+3) / (5x+7) into (2+3/x) / (5+7/x) all you have to do is divide top and bottom of (2x+3) / (5x+7) each by x.