What is the simplified form of (n+4)!/(n+1)! + (n+3)!/n!?

Correct answer: (n+3)(n+2)(2n+5)

Use factorial cancellation and expand each quotient only as far as needed. For the first term, (n+4)!/(n+1)! = (n+4)(n+3)(n+2). For the second term, (n+3)!/n! = (n+3)(n+2)(n+1). The common factor in both terms is (n+3)(n+2), so factor it out: (n+3)(n+2)[(n+4) + (n+1)]. Adding the remaining factors gives 2n + 5. Therefore the simplified form is (n+3)(n+2)(2n+5), which is option C. Option B incorrectly treats (n+4) as common to both terms, while A has the wrong final linear factor. Option D omits the correct factors from the first quotient.