Multiplying two things makes them one term.
"When a product involves a variable, it is customary to omit the symbol X of multiplication. Thus, 3 X n is written 3n and means three times n, and a X b is written ab and means a times b." Modern Algebra: Structure And Method, page 36. Immediately before the definition you're now lying about.
Fuck your non-sequitur. a(b+c)^2^ is a*(b+c)^2^, as backed up by - for example - these four math textbooks. No textbook will ever say it produces an a^2^ term. You made it up. You're just full of shit.