AMC 10/12 A Fall 2021 (Problem 17)

For how many ordered pairs

(b,c)

of positive integers does neither

x^2+bx+c=0

nor

x^2+cx+b=0

have two distinct real solutions?

\text{(A)}\;4

\text{(B)}\;6

\text{(C)}\;8

\text{(D)}\;12

\text{(E)}\;16