Unless you can see a way to just straight out factor this shiznick, I would use the quadratic formula. Which is what I did.

Just for reminder, the quadratic formula is

Code:

for the equation ax^2 + bx + c = 0
-b +/- sqrt( b^2 - 4ac)
x = -----------------------
2a
I will be using the terms "a coefficent", "b coefficent", and
"c coefficent" as general coefficents to the quadratic equation above.

Okay, with that outta tha way.

Code:

12c^2 - 4cb - 4ca + ab = 0
Start off by finding your "b coefficent".
Rewrite your equation as
12c^2 - 4(a+b)c + ab = 0

Make sense? Reply if it doesn't. I will now continue.

Refer to the above quatratic equation. Now we can say that the "a coefficent" is 12, the "b coefficent" is -4(a+b), and the "c coefficient" (not to be confused with the variable 'c' you're solving for) is ab. With that said, you can now setup the quadratic formula for your equation.

Code:

-[-4(a+b)] +/- sqrt( [-4(a+b)]^2 - 4*12*ab )
c = ----------------------------------------------
2*12
4(a+b) +/- sqrt( 16(a+b)^2 - 48ab )
c = ----------------------------------------------
24
4(a+b) +/- sqrt( **16[a^2+2ab+b^2] - 48ab** )
c = ----------------------------------------------
24
4(a+b) +/- sqrt( **16a^2 + 32ab + 16b^2 - 48ab** )
c = ----------------------------------------------
24
4(a+b) +/- sqrt( **16a^2 - 16ab + 16b^2** )
c = ----------------------------------------------
24
4(a+b) +/- sqrt( **16[a^2 - ab + b^2]** )
c = ----------------------------------------------
24
4(a+b) +/- **4*sqrt( a^2 - ab + b^2 )**
c = ----------------------------------------------
24
4
c = -- { a + b +/- sqrt( a^2 - ab + b^2 ) } Factor out 4/24
24
1
c = -- { a + b +/- sqrt( a^2 - ab + b^2 ) } Reduce. Answers.
6