If it seems too complicated, you can remove R5, R6, R8 making exact adjustment more critical.

The circuit will give you more resolution for the range you asked, -30 to +30 for a 0 to 5 volts output or about 0,083 volt/deg celsious.

Also it gives you an output for negative temperatures.

For the LM35 you will get 10mV/C. This means you will get from 0-0,3 volts for 0-30 C. If you want to measure from -30, then you must raise the ground of the LM35 above say 0,3 or better 0,6 volts. Maybe with a diode. Your output then will not be referenced to ground but from the pins of the LM35 (out and LM35 gnd). Your span for -30 to +30 will be 0,6 volts. If you need to drive an ADC then it would be better to amplify this small signal too.

HTH,
Ioannis