How do you solve this expression: 
3x^2-5x+2 can be factor as 3(x+p)^2=q


I think you mistyped.  I think you meant

How do you solve this expression: 
3x²-5x+2 can be factored as 3(x + p)² + q

Then let's multiply 3(x + p)² + q out and
see what p and q would have to be

3(x + p)² + q =
3(x + p)(x + p) + q =

3(x² + 2px + p²) + q =

3x² + 6px + 3p² + q =

Now compare that to 

3x² - 5x + 2.

Now I'll do some coloring to make it clear:

3x² + 6px + 3p² + q =

Now compare that to 

3x² - 5x + 2.

For those to be equal, those two red parts
must be equal and also those two blue parts
must be equal. IOW

          6p = -5
     3p² + q =  2

To solve that system, solve the first for p
          6p = -5 
           p = -5/6

Now substitute -5/6 for p in

     3p² + q =  2
3(-5/6)² + q = 2
3(25/36) + q = 2
   25/12 + q = 2

Clear of fractions by multiplying thru by 12

    25 + 12q = 24
         12q = -1
           q = -1/12
               
Therefore, p = -5/6 and q = -1/12. Therefore,

3x²-5x+2 can be factored as 3(x - 5/6)² - 1/12

Edwin