The value of b for which the equation $2{\log }_{1/25}(\mathrm{bx}+28)=-{\log }_5\left(12-4\mathrm{x}-{\mathrm{x}}^2\right)$ has coincident roots is

$\begin{array}{l}\frac{2{\log }_5(\mathrm{bx}+28)}{{\log }_5(1/5{)}^2}=-{\log }_5\left(12-4\mathrm{x}-{\mathrm{x}}^2\right)\\ \Rightarrow \mathrm{bx}+28=12-4\mathrm{x}-{\mathrm{x}}^2\\ \text{ or }{\mathrm{x}}^2+(\mathrm{b}+4)\mathrm{x}+16=0\end{array}$

For coincident roots,

$\begin{array}{l}\mathrm{D}=0\\ \Rightarrow (\mathrm{b}+4{)}^2=4(16)\\ \Rightarrow \mathrm{b}+4=\pm 8\end{array}$