Proof by simple induction, question concerning the inductive step

Question

Sorry in advance for posting a picture instead of the text, I'm not familiar with Latex yet...

My question is...

Why do we get: 4n^2 + 12n + 7 = 4((n − 1) + 1)^2 + 12((n − 1) + 1) + 7

And not this: 4n^2 + 12n + 7 = 4(n+1)^2 + 12(n+1) + 7

Proof

Answer 1

The induction step should be "Assume that $P(n-1)$ holds, and then we show that $P(n)$ holds". This is an inaccuracy in the proof. Also, you should never say "Suppose we know that $P(n)$ is true", just say "Suppose that $P(n)$ is true". (That is the induction hypothesis, not the fact that we know it...)