Refer to the diagram of the circle inscribed in a

quadrilateral ABCD with the points of tangency at M, N, P,

and Q. For each positive integer n, a diagram is drawn like

this in which lengths of line segments are

AQ = AM = n,

BM = BN = n + 1,

CN = CP = n + 2, and

DP = DQ = n + 3.

The radius of the circle is $\displaystyle \sqrt{f (n)} $, where$\displaystyle f (n) $ is a

polynomial function in n.

Find $\displaystyle f (n) $