A rectangle has length (r+6) and breadth (r+2). Which expansion is obtained from the area model?

Correct answer: r^2+8r+12

The direct answer is A: r^2+8r+12. The area of the rectangle is length multiplied by breadth, so calculate (r+6)(r+2). In an area model, split each side into its variable and constant parts. The four regions have areas r×r=r^2, r×2=2r, 6×r=6r, and 6×2=12. Adding them gives r^2+2r+6r+12=r^2+8r+12. Therefore A is correct. B incorrectly uses 12r as the middle term; 12 is the product of the constants and must be the constant term, not a coefficient of r. C adds the constant terms instead of combining the two variable rectangles. D has the right middle coefficient but the wrong constant product: 6×2 is 12, not 8. A useful check is to multiply symbolically: (r+6)(r+2)=r^2+2r+6r+12. In a rectangle model, every one of the four smaller regions must be counted exactly once.