Let m\in Z such that

n^2+9n+14=m^2,

A perfect square number can be expressed by a smaller perfect number plus an integer, which can be further expressed as the product of two inters using the difference of two squares.

4n^2+36n+4\times 14=4m^2

4n^2+36n+81+4\times 14=4m^2+81

(2n+9)^2-4m^2=81-56 = 25

(2n+9-2m)(2n+9-2m) = 25

Let a = 2n+9-2m, b = 2n+9-2m

Addition of above equations gives

n = \dfrac{a+b-18}{4}

and

a\cdotp b =1\times 25=5\times 5 = 25

When a = 1, b=25,

n =\dfrac{26-18}{4} = 2

n^2+9n+14 = 2^2+9\times 2+14=36

m = 6

When

So B is the choice