Remember when dealing with a constraint function you have to remove the constants and create a new function g(x,y) (basically, the entire left hand side of the equation) that consists of all the variable terms of the function, giving you:
Making your gradient:
You know that:
Which gives you a system of equations:
3xy + 4xz + 4yz = 2C
Solve for lambda first, then get equations in terms of JUST x,y, and z, and solve for each simultaneously (Unless solving for lambda gives you some clue as to the solution of 1 or more variables) and you will get the critical points.