Groups of terms beside each other are multiplying each other

Actually the whole thing is a single Term. Terms are separated by operators and joined by grouping symbols, and there's no operators between the successive brackets.

you have to solve what’s inside of those groups before multiplying them together

You don't have to, but it sure makes the working-out a lot easier if you do!

If a=1, b=2, c=3, d=4, then...

(a+b)(c+d)=(ac+ad+bc+bd)

(1+2)(3+4)=(1x3+1x4+2x3+2x4)=(3+4+6+8)=21

whereas...

(1+2)(3+4)=(3)(7)=(3x7)=21 :-)