Factoring this expression..

Can someone please help me understand how to get from this:
3pv - p + 9/v - 3/v² = 8t

To this:
(p + 3/v²)(v - 1/3) = 8t/3

How do you factor when you have things like 1/v², etc..? Please show steps and explain, thank you!

3pv - p + 9/v - 3/v² = 8t
First take out p as a common factor
p(3v - 1) + 9/v - 3/v² = 8t
Now take out 3/v²
p(3v - 1) + 3/v²(3v - 1) = 8t
Now taking (3v-1) as a common factor
(3v-1) (p + 3/v²) = 8t
Finally take 3 out of the first bracket
3(v - 1/3)(p + 3/v²) = 8t
(v - 1/3)(p + 3/v²) = 8t/3

I don't know what the easiest or best way would be, but this is one way to do it:
You could divide the 1st term by the 2nd term, and then the 3rd term by the 4th term, and see if they're the same. If they're the same, you can factor the expression.

In your example, you could see if
3pv . . . . 9/v
----- and ------- are the same.
­ -p . . . . -3/v²

You get -3v in both cases, so they're the same.
This means the 1st term is (-3v) times the 2nd term: . 3pv = (-3v)×(-p)
and the 3rd term is (-3v) times the 4th term: . . . . . . . 9/v = (-3v)×(-3/v²)

So you can write the equation like this:
(-3v)×(-p) + 1×(-p) + (-3v)×(-3/v²) + 1×(-3/v²) = 8t
Now the first two terms can be factored to (-3v+1)(-p),
and the next two terms can be factored to (-3v+1)(-3/v²).

So you get:
(-3v+1)(-p) + (-3v+1)(-3/v²) = 8t
Factor again:
(-3v + 1)(-p - 3/v²) = 8t
Do a few adjustments:
-3(v - 1/3)(-p - 3/v²) = 8t
-(v - 1/3)(-p - 3/v²) = 8t/3
-(-p - 3/v²)(v - 1/3) = 8t/3
(p + 3/v²)(v - 1/3) = 8t/3