If in the rectangle model of (x+a)(x+b), ab = 42 and the sum of the middle strips is 13x, what can a and b be?

Correct answer: 6 and 7

The direct answer is D: a=6 and b=7, in either order. Expand the rectangle model: (x+a)(x+b)=x²+(a+b)x+ab. The constant corner requires ab=42, while the two middle strips are ax and bx, whose sum is (a+b)x=13x. So we need two numbers with product 42 and sum 13. The factor pairs of 42 are 1 and 42, 2 and 21, 3 and 14, and 6 and 7. Only 6+7=13, so D satisfies both conditions. A has the correct product but sum 43. B has product 42 but sum 17. C has product 42 but sum 23. The key lesson is to use both the corner tile and the middle strips; matching only the product can lead to a wrong choice. Since multiplication is commutative, the order of 6 and 7 does not matter.