A Sum as Input

When the input is a sum, the brackets are what keep the work honest. Take g(x) = x^2 + 2x and the input k - 1. Using the rule gives (k - 1)^2 + 2(k - 1): every x became the whole of k - 1.

Now expand each part. (k - 1)^2 is k^2 - 2k + 1, and 2(k - 1) is 2k - 2. Writing them in one line gives k^2 - 2k + 1 + 2k - 2, and collecting like terms leaves k^2 - 1.

The common slip is squaring each part on its own: writing k^2 + 1 for (k - 1)^2. The square of a sum has a middle term: twice the product of its two parts.