Alan

As follows:

Edit:  The third line in my derivation should say AD² not AO².  The numbers are correct though.

"Find the largest positive integer n such that (n^2 + 7n + 12)/(n^2 - n - 12) is an integer."

As follows:

No! It doesn't:

\(\frac{3992004}{3988009}=1.001001753005071...\\...\\ \frac{3997}{3993}=1.0010017530678688...\)

Close, but no cigar!

As follows (using the Feynman technique):

See 3D graph:

Oops!  I was careless in looking at the graph and misled myself (and you!).   The point where the two functions meet at a single point is at x = 48, not 50.  Plug this into the first function to get y (= 114).  Then use this value of y in the second function, together with x = 48 to get k ( =  18).

I answered this yesterday - see: https://web2.0calc.com/questions/please-help-how-do-you-graph-parametric-equations#r1 

Oops! Thank you, you are correct. I've kept a 5! in the denominator, after I'd cancelled it with part of the 10! in the numerator.  I'll change it.

As follows:

\(\frac{10!}{2!(10-2)!}\times \frac{8!}{3!(8-3)!}= \frac{10!}{2!}\times \frac{1}{3!5!}=\frac{10*9*8*7*6}{2*6}=2520\)

Edited to correct silly error!