Polynomial distribution and a check

A learner multiplies the supplied factors (x+3) and (x+2) and writes x-squared plus six. Another adds the two middle products without keeping their variable factors. A third draws a verbal area model with four regions: x by x, x by two, three by x and three by two. The task asks for the expanded polynomial and a numerical check at x=2. This is an algebraic model with explicit factors; it does not require an unseen diagram or information about an actual room. The response should explain why every term in one factor multiplies every term in the other, and why only like terms combine. A correct substitution is a useful check of a proposed expression, although one matching value alone would not prove every possible polynomial identity.