What is the simplified form of \(\frac{(n+2)!-(n+1)!}{n!}\)?

Correct answer: \((n+1)^2\)

Using \((n+2)!=(n+2)(n+1)!\), the numerator becomes \((n+1)![(n+2)-1]=(n+1)(n+1)!\). Now, \((n+1)!=(n+1)n!\), so the numerator is \((n+1)^2n!\). Dividing by \(n!\) gives \((n+1)^2\). The expression \(n(n+1)\) would result from the incorrect step \((n+1)!=n\cdot n!\). Exam tip: rewrite factorial terms in terms of a common smaller factorial, such as \(n!\), before simplifying.