Algebra

Notice that $x^2 + y^2 = 4x + 5y$ is very similar to the equation of a circle. 

We have $x^2-4x+y^2-5y=0$. 

Now, completing the square for both x and y, we get

$(x-2)^2+(y-5/2)^2=25/4+16$

This is the official equation of a circle. 

It has the center at $(2, 5/2)$ and a radius of $\sqrt{89/4}$

The largest value of x is basically the x value of the center added to the radius. 

Thus, we have $2+\frac{\sqrt{89}}{2} = \frac{4+\sqrt{89}}{2}$

So our final answer is $\frac{4+\sqrt{89}}{2}$

Thanks! :)