if you see a number butted up against an expression in parentheses you assume there is a multiplication symbol there

No, it means it’s a Term (product). If a=2 and b=3, then axb=2x3, but ab=6.

I was taught 2(2+2)==2*(2+2)

2(2+2)==(2*(2+2)). More precisely, The Distributive Law says that 2(2+2)=(2x2+2x2).