To shut up the Argies ones and for all about Falkland claims. It seems that Argentina wants $$$ to revive their declining economy that's why they want to claim Falklands. The win-win solution that I could see is that Argentina should renounce its claims in Falkland in exchange of British investment in Argentina then Argentina receives a part $$$ from oil drilling and Argentina transforms itself into a major oil producer and becomes the richest country in the world with the help of Britain and the rest of western nations. The oil revenues should be divided by 50-50 proportion, 50% goes to Argentina and 50% goes to Britain. Had Argentina used their balls, their economic problems will be fixed ones and for all.
They did use their balls. That's why they attacked (hoping, apparently, for the US to pressure Britain to accept the fait accompli). Without oil (or control of southern ocean fisheries for which the Brits settled them in the first place), the Falklands are economically worthless. Argentina wanted them for pride.
You mean they should have used their heads. As others have said, the British had no reason to work with them.