To solve this system of equations, we can first solve the first equation for y in terms of x:

4y + x = 0
4y = -x
y = -x/4

Next, substitute this expression for y into the second equation:

x^2 + (-x/4)^2 = 17
x^2 + x^2/16 = 17
16x^2 + x^2 = 272
17x^2 = 272
x^2 = 16
x = ±4

Now that we have found the possible values for x, we can substitute them back into the equation we found for y:

If x = 4:
y = -(4)/4
y = -1

If x = -4:
y = -(-4)/4
y = 1

Therefore, the solutions to the system of equations are (4, -1) and (-4, 1).