In a tile model there is one \(x^2\) tile and eight \(x\)-tiles. If there are 16 constant tiles, what complete square does the model represent?

Correct answer: \((x+4)^2\)

The direct answer is A: \((x+4)^2\). The tiles represent the expression \(x^2+8x+16\). To recognize a perfect square, compare it with \((x+c)^2=x^2+2cx+c^2\). Since the constant term is 16, we take \(c=4\), because \(4^2=16\). Then the middle term should be \(2\cdot4\cdot x=8x\), exactly as shown by the eight \(x\)-tiles. Therefore \(x^2+8x+16=(x+4)^2\). B is wrong because \((x+8)^2=x^2+16x+64\), not the given expression. C would produce \(x^2-8x+16\), with a negative middle term. D leaves the expression ungrouped and omits the eight \(x\)-tiles. Memory cue: half the coefficient of \(x\) gives the added number; half of 8 is 4, and its square is 16.