I have a Dual core processor that get hot while multitasking and using some 'heavy' softwares. How keep it cool without adding additional cooling hardware?
Keep it clean. Clean dust off computer parts. Compressed air works great for this.
Good airflow. Make sure your computer has good airflow around the case. Don't pile up stuff around your computer and make sure all air inlets and outlets are free and clear.
Cable management. If your computer has a lot of various cable jammed in it running all over the place, you might take a few minutes and try and clean up the cable organization.
Thermal Paste. If the machine is older and the CPU thermal paste is dirty/dry, it may be worth your efforts to pull the heatsink off the processor and re apply some thermal paste. Make sure to clean off all old thermal paste before applying the new. Not sure if you didn't mind spending a few bucks, but Ryan (in comments) brought up this cheap option.
Does the computer have existing fans in it that can be moved around to better locations? If so, you might be able to figure out a better cooling layout. Are all your current fans currently running ok? Beyond that, it is replacing or adding extra cooling hardware if you want to cool further.
One word: Undervolting
Undervolting reduces the power consumed by the processor by lowering its input voltage to a lower than factory level. Less power = less temperature.
Undervolting does not decrease performance and is completely safe as lowering the voltage cannot harm your hardware. It even increases your battery life, as less power is drained from the battery.