Students will prove Sophie Germain's identity. One way of proving Sophie Germain's identity is easy, as it requires nothing more than distributing and combining like terms. When we ask students to prove the identity, this is the route we expect them to take. However, there is another way by completing the square, followed by factoring as a difference of squares. It's unlikely that students would think of this way, so we'll just show it to them. Hopefully this gets them to realize completing the square and factoring differences of squares apply to a wider variety of problems than students previously thought.
After students have proven the theorem themselves, and seen the alternative proof, give your students this problem.
This ProofWiki page, unfortunately, has a mistake in the first proof of Germain's identity. The other variation on the ProofWiki page is correct, but backwards, so I wouldn't recommend teaching it.
Should I include this problem?
Conclude by giving your students these challenges: