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

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 $\boxed{{\displaystyle (12,-4)}}$ and $\boxed{{\displaystyle (0,2)}}$.