AMC 10 2021-fall-a Problem 20

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? $\textbf{(A) } 4 \textbf{(B) } 6 \textbf{(C) } 8 \textbf{(D) } 12 \textbf{(E) } 16$