Solve the non-linear system of equations.
\[
\begin{array}{l}
(x-4)^{2}+(y+5)^{2}=65 \\
x+2 y=4
\end{array}
\]

Step 1 ：Solve the non-linear system of equations.

Step 2 ：The first equation represents a circle with center \((4, -5)\) and radius \(\sqrt{65}\).

Step 3 ：The second equation represents a straight line.

Step 4 ：To find the intersection points, substitute the expression for \(x\) from the second equation into the first equation and solve for \(y\).

Step 5 ：Then, use the value of \(y\) to find \(x\).

Step 6 ：Substitute \(x = 4 - 2y\) into the first equation to get \(4y^2 + (y + 5)^2 = 65\).

Step 7 ：Solve for \(y\) to get the solutions \(y = -4\) and \(y = 2\).

Step 8 ：Using these \(y\) values, find the corresponding \(x\) values to get the solutions \((12, -4)\) and \((0, 2)\).

Step 9 ：The solutions to the system of equations are \(\boxed{(12, -4)}\) and \(\boxed{(0, 2)}\).