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Well, they thoughtthey destoyed the suit, all but a little piece was actually destroyed. Since Eddie Brock's(i think) dad worked in the lab on the suit, he was the one that wanted to keep a piece of his life's work and ended up passing it on to his son. That is how symbiote know as Venom was created, according to the special on G4.
Pretty close. Before I go further I must make this Public Service Announcement:
Read Ultimate Spider Man. Best.Spider-man.Ever.
On topic, in the Ultimate storyline, it was Brock's dad as well as Peter Parker's who were working on this black substance. Brock and Peter have a different relationship than you'd normally expect. I won't tell you here because for those who haven't read, it sounds like blasphemy. But trust me, read a couple volumes in the graphic novel section, and it'll make total sense.
I can't really take the 'scooper's' word for it about what Venom looks like, since they are no pictures. Plus I liked the designs of the original, whereas they've come to some consensus that the designs for the first one sucked. As Ian mentioned earlier, it'd be goofy if the Green Goblin was just a rubber suit.
Posssible idea to introduce Venom, movie style
Eddie Brock is a fellow student in Connor's class
Connor's is working on black substance
Parker has an 'accident' with said substance
Black Spider Man does his thing, until it becomes too much
The symbiote seperates, and by another accident, Brock becomes Venom
A bit rough, and maybe a bit goofy, but with a little tweaking, it could work...