Do you need to measure both + and - voltages on the same input? In other words does the signal swing from +2V to -2V?

If so, then you must use an op-amp (LM324 or equivalent) and offset the input to the op-amp by 2V. So -2V now becomes 0V and +2V becomes +4V.
In software you would compensate by subtracting 2V (half scale of A/D = 7FF) from the A/D result.

Jacob,

if you apply +2V to your circuit, at non-inverting input of LM324 also +2V will be found. So, you need a gain of +2 to produce +4V at output of LM324.

If you apply 0V at input of circuit, at non-inverting input of LM324 +1V will be found. So, also gain of +2 is needed to produce wished +2V at output of LM324.

You just forgot, that those two 10kOhm resistors at input not only provide a bias of +2V, but also a dividing of input voltage by a factor of two.

Kai