You want to power a laptop computer for 8 hours a day.
An average laptop typically consumes around 60W. So 60W X 8 = 480 Wh
Double this amount to avoid over-discharging of battery and extend it’s life. So 480Wh X 2 = 960Wh of power every day needed.
A 12V, 80Ah deep-cycle, lead-acid battery is more than enough. This battery has a 12V X 90Ah = 1080Wh capacity.

Let’s say there’s an average of 4 hours of sunshine every day, so we need to produce these 1080Wh during every day. This means our solar panel has to be 1080Wh/4h = 270W. In conclusion, you’re going to need:

170 W solar panel
110 W solar panel
Inverter 12V to 110V (220V) 100W (Or just a 12V to 19V DC converter)

There will probably be even more surplus energy, especially during summer time, so you this system can deliver more power during summer.

A 12V to 19V DC converter, especialy designed to power laptops is probably a better choice for the laptop. On the other hand, a 110V AC (220V AC) inverter can power a host of other devices, (lamps, mini-stereos, small TVs, radios, various chargers, etc) and is a good thing to have around the house in case of emergency. For example, during power outage, you can hook the inverter to the car battery and use a long cable to carry 110V ( 220V) AC power inside your home.

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