That's a circle with center (-2,5) and radius 2 intersecting a line with
slope 4 and y-intercept k. So, k, the y-intercept of the line will be the
largest when the line is as far to the left of the circle and still intersects
the circle.



That's when the line is tangent to the circle on the left side. That's also 
when the distance from the line to the center is the radius 2.

We use the point-to-line distance formula:

We get 0 on the right side of the line's equation:



Perpendicular distance from the point (x1,y1)
to the line Ax+By+C=0 is













We want k to be a maximum so we use the + sign.

 approximately 21.24621125.

Oh darn, I just realized that x and y had to be integers. Instead of
starting over it looks like the nearest integer x could be -4. So we have
to move the line a tiny bit right.

So, we plug -4 for x in the circle equation:







So (-4,5) is the nearest point to the point of tangency that
has both coordinates integers.
So the line y=4x+k should go through (-4,5), moving it a tiny
bit right.






So the answer is 21.
 
Edwin

Ikleyn is apparently not familiar with the standard 
notation of Z for the set of all integers.  I had
forgotten it myself and did not notice it. But the
solution by MathLover1 is the correct one. Mine at
the top is correct for the actual maximum value of
k but required some "eyeballing" to find the actual
maximum, assuming x and y to be integers.  Copied
from the internet:

The set of integers is represented by the letter Z. 
An integer is any number in the infinite set,

Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Integers are sometimes split into 3 subsets, 

,  and {0}.  is the 
set of all positive integers (1, 2, 3, ...), while 
is the set of all negative integers (..., -3, -2, -1). 
Zero is not included in either of these sets .  
is the set of all positive integers and 0, while  
is the set of all negative integers and 0. 

Edwin