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I'm assuming the constraint involves some plus signs that aren't appearing for some reason, so that you're finding the extrema subject to 
.
Set
and 
, so that the Lagrangian is
Take the partial derivatives and set them equal to zero.
One way to find the possible critical points is to multiply the first three equations by the variable that is missing in the first term and dividing by 2. This gives
So by adding the first three equations together, you end up with
and the fourth equation allows you to write
Now, substituting this into the first three equations in the most recent system yields
So we found a grand total of 8 possible critical points. Evaluating at each of these points, you find that 
attains a maximum value of 
whenever exactly none or two of the critical points' coordinates are negative (four cases of this), and a minimum value of 
whenever exactly one or all of the critical points' coordinates are negative.
Set
Take the partial derivatives and set them equal to zero.
One way to find the possible critical points is to multiply the first three equations by the variable that is missing in the first term and dividing by 2. This gives
So by adding the first three equations together, you end up with
and the fourth equation allows you to write
Now, substituting this into the first three equations in the most recent system yields
So we found a grand total of 8 possible critical points. Evaluating