Example 1 Given that $a$ and $b$ are the roots of the equation
$$
x^{2}-4 x+m=0
$$

and $b$ and $c$ are the roots of the equation
$$
x^{2}-8 x+5 m=0
$$

then $m=$ $\qquad$
Given that $b$ is a common root of the two given equations, we have
$$
\begin{array}{l}
b^{2}-4 b+m=0, \\
b^{2}-8 b+5 m=0 .
\end{array}
$$

From equation (1), we get $m=4 b-b^{2}$.
Substituting this into equation (2) and simplifying, we get $b^{2}-3 b=0$.
Solving this, we find $b=0$ or 3.
Therefore, $m=4 b-b^{2}=0$ or 3.