If an \(a\) by \(a\) square is joined with an \(a\) by \(b\) rectangle, how can the formed area be written?

Correct answer: \(a(a+b)\)

The direct answer is option A. The square contributes area \(a\times a=a^2\), and the rectangle contributes area \(a\times b=ab\). Adding them gives \(a^2+ab\). Both terms contain the common factor \(a\), so factorising gives \(a(a+b)\). Geometrically, the two pieces share the same width \(a\); their lengths \(a\) and \(b\) combine to make a total length \(a+b\). Therefore the joined shape has area \(a(a+b)\). Option B is only a length, not an area. Option C incorrectly treats the sum as two separate squares and omits the mixed product. Option D has the wrong powers and does not equal \(a^2+ab\) in general. Memory cue: common width times combined length gives the factored form.