It is given that \(2+4 i\) is one of the roots of \(x^2+b x+c=0\) with \(b, c \in \mathbf{R}\), so other root will be \(2-4 i\).
Now, the sum of roots \(=-b\)
\(\Rightarrow \quad(2+4 i)+(2-4 i)=-b \Rightarrow b=-4\)
and the product of roots \(=c\)
\(\begin{aligned}
& \Rightarrow \quad(2+4 i)(2-4 i)=c \\
& \Rightarrow \quad 4+16=c \Rightarrow c=20 \\
& \therefore \quad(b, c)=(-4,20) \\
\end{aligned}\)
Hence, option (d) is correct.